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Beware of bugs in the above code; I have only proved it correct, not tried it. -- Donald Knuth

tech / sci.math / Re: Natural numbers and vases III

SubjectAuthor
* Natural numbers and vases IIIWM
+* Re: Natural numbers and vases IIIFritz Feldhase
|+- Re: Natural numbers and vases IIIGus Gassmann
|`* Re: Natural numbers and vases IIIWM
| `- Re: Natural numbers and vases IIIsergi o
+* Re: Natural numbers and vases IIIsergi o
|`* Re: Natural numbers and vases IIIFromTheRafters
| +- Re: Natural numbers and vases IIIsergi o
| `* Re: Natural numbers and vases IIIWM
|  `* Re: Natural numbers and vases IIIsergi o
|   `- Re: Natural numbers and vases IIIsergi o
+* Re: Natural numbers and vases IIIWilliam
|`* Re: Natural numbers and vases IIIWM
| `* Re: Natural numbers and vases IIIWilliam
|  `* Re: Natural numbers and vases IIIWM
|   `* Re: Natural numbers and vases IIIWilliam
|    `* Re: Natural numbers and vases IIIWM
|     `* Re: Natural numbers and vases IIIWilliam
|      +* Re: Natural numbers and vases IIIWM
|      |`* Re: Natural numbers and vases IIIWilliam
|      | `* Re: Natural numbers and vases IIIWM
|      |  +* Re: Natural numbers and vases IIIWilliam
|      |  |+- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |+- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |+* Re: Natural numbers and vases IIIFritz Feldhase
|      |  ||`* Re: Natural numbers and vases IIIWM
|      |  || `* Re: Natural numbers and vases IIIsergi o
|      |  ||  `* Re: Natural numbers and vases IIIEram semper recta
|      |  ||   `- Re: Natural numbers and vases IIIEram semper recta
|      |  |`* Re: Natural numbers and vases IIIWM
|      |  | +- Re: Natural numbers and vases IIIsergi o
|      |  | `* Re: Natural numbers and vases IIIWilliam
|      |  |  +* Re: Natural numbers and vases IIIFritz Feldhase
|      |  |  |`* Re: Natural numbers and vases IIIWM
|      |  |  | +- Re: Natural numbers and vases IIIFromTheRafters
|      |  |  | `* Re: Natural numbers and vases IIIFritz Feldhase
|      |  |  |  +* Re: Natural numbers and vases IIIWM
|      |  |  |  |`- Re: Natural numbers and vases IIIFromTheRafters
|      |  |  |  +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |  |  `- Re: Natural numbers and vases IIIGus Gassmann
|      |  |  `* Re: Natural numbers and vases IIIWM
|      |  |   +* Re: Natural numbers and vases IIIWilliam
|      |  |   |`* Re: Natural numbers and vases IIIWM
|      |  |   | +* Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | ||`- Re: Natural numbers and vases IIIFromTheRafters
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | ||+* Re: Natural numbers and vases IIIFromTheRafters
|      |  |   | |||`- Re: Natural numbers and vases IIISurgio
|      |  |   | ||`* Re: Natural numbers and vases IIIJim Burns
|      |  |   | || +* Re: Natural numbers and vases IIIWM
|      |  |   | || |+- Re: Natural numbers and vases IIISergio
|      |  |   | || |`* Re: Natural numbers and vases IIIJim Burns
|      |  |   | || | `* Re: Natural numbers and vases IIIWM
|      |  |   | || |  `* Re: Natural numbers and vases IIISergio
|      |  |   | || |   `- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | || +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | || +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | || +* Re: Natural numbers and vases IIIWM
|      |  |   | || |`- Re: Natural numbers and vases IIISergio
|      |  |   | || +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | || `- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | ||`- Re: Natural numbers and vases IIISurgio
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | ||`- Re: Natural numbers and vases IIISurgio
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | ||`- Re: Natural numbers and vases IIISergio
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | ||`- Re: Natural numbers and vases IIISergio
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | ||`- Re: Natural numbers and vases IIISergio
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | ||`- Re: Natural numbers and vases IIISergio
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | ||`* Re: Natural numbers and vases IIISergio
|      |  |   | || +* Re: Natural numbers and vases IIIWM
|      |  |   | || |`* Re: Natural numbers and vases IIISergio
|      |  |   | || | +* Re: Natural numbers and vases IIIWM
|      |  |   | || | |`* Re: Natural numbers and vases IIISergio
|      |  |   | || | | `- Re: Natural numbers and vases IIIJim Africani
|      |  |   | || | `- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | || `- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | ||`- Re: Natural numbers and vases IIISergio
|      |  |   | |+* Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIFredJeffries
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIFredJeffries
|      |  |   | |+* Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIFredJeffries
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWilliam
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+- Re: Natural numbers and vases IIIWM
|      |  |   | |+* Re: Natural numbers and vases IIIWM
|      |  |   | |`* Re: Natural numbers and vases IIIWM
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +* Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +* Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +* Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIEram semper recta
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +* Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +* Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +* Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIEram semper recta
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +* Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +* Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIEram semper recta
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIEram semper recta
|      |  |   | +- Re: Natural numbers and vases IIIEram semper recta
|      |  |   | +* Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIEram semper recta
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIEram semper recta
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +* Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +- Re: Natural numbers and vases IIIGus Gassmann
|      |  |   | +* Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   | `- Re: Natural numbers and vases IIIzelos...@gmail.com
|      |  |   `* Re: Natural numbers and vases IIIFromTheRafters
|      |  `* Re: Natural numbers and vases IIIzelos...@gmail.com
|      `* Re: Natural numbers and vases IIIFritz Feldhase
+* Re: Natural numbers and vases IIIzelos...@gmail.com
+* Re: Natural numbers and vases IIIEram semper recta
+- Re: Natural numbers and vases IIIEduardo Faqtardo
+- Re: Natural numbers and vases IIIEram semper recta
+- Re: Natural numbers and vases IIIArchimedes Plutonium
`- Re: Natural numbers and vases IIIArchimedes Plutonium

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Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: franz.fr...@gmail.com (Fritz Feldhase)
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 by: Fritz Feldhase - Tue, 16 Aug 2022 18:16 UTC

On Tuesday, August 16, 2022 at 5:39:08 PM UTC+2, WM wrote:

> The [natural] numbers cannot leave the finite domain.

Indeed! And t h e y don't.

But the NUMBER of natural numbers can and does.

Hint: There are infinitely many natural numbers.

> Proof: {1, 2, 3, ..., n} is finite for every [natrural] number n

Indeed!

And the set {1, 2, 3, ...} = {n : n e IN} of _all_ natural numbers is _infinite_.

Moreover: card(1, 2, 3, ...}) = aleph_0 !e IN.

You see:

1. An e IN: {1, 2, 3, ..., n} is finite & card({1, 2, 3, ..., n}) e IN

2. {1, 2, 3, ...} = {n : n e IN} is infinite & card({1, 2, 3, ...}) !e IN

(since card({1, 2, 3, ...}) = aleph_0).

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: zelos.ma...@gmail.com (zelos...@gmail.com)
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 by: zelos...@gmail.com - Wed, 17 Aug 2022 05:16 UTC

tisdag 16 augusti 2022 kl. 17:39:08 UTC+2 skrev WM:
> William schrieb am Dienstag, 16. August 2022 um 16:50:08 UTC+2:
> > On Tuesday, August 16, 2022 at 7:10:30 AM UTC-3, WM wrote:
> > > William schrieb am Montag, 15. August 2022 um 22:02:03 UTC+2:
> > > > On Monday, August 15, 2022 at 4:09:52 PM UTC-3, WM wrote:
> > > >
> > > > > Subtract all.
> > > >
> > > > Ok, Subtract the set.
> > > The set is all its elements.Therefore subtract the set means subtract all its elements.
> > > > >Then you can say: I have subtracted all Peano numbers
> > > > "All Peano Numbers" is the set of Peano numbers". This set contains S(n) for any element n. So if you remove the set you remove each S(n).
> > > And if you remove each S(n) then nothing remains.
> > Correct. If you "subtract all" then nothing remains. So your contention
> >
> > > Subtract all. Then you can say: I have subtracted all Peano numbers but most Peano numbers remain as successors
> >
> > is false.
> No. Individual subtraction leaves all dark numers.

There are no dark numbers!

> > There is no contradiction. "all n which belong to a Peano set and have ℵo successors" is one set
> No, if so, then you could remove them by induction, individually. But you can't. The successors form an ℵ₀-set. The Peano numbers cannot leave the finite domain.
> Proof: {1, 2, 3, ..., n} is finite for every Peano number, for every definable Peano number n.

That proves NOTHING of what you want!

>
> Regards, wm

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
Injection-Date: Wed, 17 Aug 2022 14:46:30 +0000
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 by: WM - Wed, 17 Aug 2022 14:46 UTC

Fritz Feldhase schrieb am Dienstag, 16. August 2022 um 20:16:32 UTC+2:
> On Tuesday, August 16, 2022 at 5:39:08 PM UTC+2, WM wrote:
>
> > The [natural] numbers cannot leave the finite domain.
>
> Indeed! And t h e y don't.
>
> But the NUMBER of natural numbers can and does.

The number of natural numbers is a natural number.
>
> Hint: There are infinitely many natural numbers.

Hint: That is potential infinity: Increasing without halt, but always remaining finite.
>
> > Proof: {1, 2, 3, ..., n} is finite for every [natrural] number n
>
> Indeed!
>
> And the set {1, 2, 3, ...} = {n : n e IN} of _all_ natural numbers is _infinite_.

You need to refrain from individually describing the elements.
>
> Moreover: card(1, 2, 3, ...}) = aleph_0 !e IN.
>
> You see:

Only with dark numbers the set can be handled.

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
Injection-Date: Wed, 17 Aug 2022 14:51:28 +0000
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 by: WM - Wed, 17 Aug 2022 14:51 UTC

William schrieb am Dienstag, 16. August 2022 um 19:25:41 UTC+2:
> On Tuesday, August 16, 2022 at 12:39:08 PM UTC-3, WM wrote:
> > William schrieb am Dienstag, 16. August 2022 um 16:50:08 UTC+2:
> > > On Tuesday, August 16, 2022 at 7:10:30 AM UTC-3, WM wrote:
>
> > > > And if you remove each S(n) then nothing remains.
> > > Correct. If you "subtract all" then nothing remains. So your contention
> > >
> > > > Subtract all. Then you can say: I have subtracted all Peano numbers but most Peano numbers remain as successors
> > >
> > > is false.
> > No. Individual subtraction leaves all dark numers.
> Vacuously true. There are no "dark numbers"

There are numbers which can only be removed by removing the set: ℵo successors.

> > > There is no contradiction. "all n which belong to a Peano set and have ℵo successors" is one set
> > No, if so, then you could remove them by [a stepwise process]. But you can't.
>
> Nope. You can remove the set "all n which belong to a Peano set and have ℵo successors" but not by a stepwise
> process.

All definable numbers can appear in a stepwise process. Dark numbers cannot..

> > The successors form an ℵ₀-set.
> Correct. The set of successors is infinite.

They are never removed stepwise.

> >The Peano numbers cannot leave the finite domain.
> > Proof: {1, 2, 3, ..., n} is finite for every Peano number,
> So what? Each *element* of the set of successors is finite. The *set* of successors is infinite

and dark.

Regards, WM

Re: Natural numbers and vases III

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From: FTR...@nomail.afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Wed, 17 Aug 2022 11:32:21 -0400
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 by: FromTheRafters - Wed, 17 Aug 2022 15:32 UTC

WM wrote on 8/17/2022 :
> Fritz Feldhase schrieb am Dienstag, 16. August 2022 um 20:16:32 UTC+2:
>> On Tuesday, August 16, 2022 at 5:39:08 PM UTC+2, WM wrote:
>>
>>> The [natural] numbers cannot leave the finite domain.
>>
>> Indeed! And t h e y don't.
>>
>> But the NUMBER of natural numbers can and does.
>
> The number of natural numbers is a natural number.

Good one, tell me another. This time I want unicorns aplenty.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: wpihug...@gmail.com (William)
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 by: William - Wed, 17 Aug 2022 15:57 UTC

On Wednesday, August 17, 2022 at 11:51:34 AM UTC-3, WM wrote:

> > >The Peano numbers cannot leave the finite domain.
> > > Proof: {1, 2, 3, ..., n} is finite for every Peano number,
> > So what? Each *element* of the set of successors is finite. The *set* of successors is infinite
> and dark.

Piffle. Each *element* or the set of successors is not "dark" and can be subtracted by a stepwise process. The fact that the *set* of successors cannnot be subtracted by a stepwise process does not mean there is an *element* of the set of successors that cannot be subtracted by a stepwise process.

--
William Hughes

Re: Natural numbers and vases III

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From: inva...@invalid.com (Sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Wed, 17 Aug 2022 11:12:59 -0500
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 by: Sergio - Wed, 17 Aug 2022 16:12 UTC

On 8/17/2022 9:46 AM, WM wrote:
> Fritz Feldhase schrieb am Dienstag, 16. August 2022 um 20:16:32 UTC+2:
>> On Tuesday, August 16, 2022 at 5:39:08 PM UTC+2, WM wrote:
>>
>>> The [natural] numbers cannot leave the finite domain.
>>
>> Indeed! And t h e y don't.
>>
>> But the NUMBER of natural numbers can and does.
>
> The number of natural numbers is a natural number.

the corollary: the natural of a number natural is a number natural.

>>
>> Hint: There are infinitely many natural numbers.
>
> Hint: That is potential infinity: Increasing without halt, but always remaining finite.

Hint: Finite remaining halts increasing infinitely potentially.

>>
>>> Proof: {1, 2, 3, ..., n} is finite for every [natrural] number n
>>
>> Indeed!
>>
>> And the set {1, 2, 3, ...} = {n : n e IN} of _all_ natural numbers is _infinite_.
>
> You need to refrain from individually describing the elements.

Translation: "Do not disagree with Me."

>>
>> Moreover: card(1, 2, 3, ...}) = aleph_0 !e IN.
>>
>> You see:
>
> Only with dark numbers the set can be handled.

Unlike Aether, which has much greater value, dark numbers are a non-existing idea held by 1 or 2 people to schmooze over on things, to hide bad math.

>
> Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: horand.g...@gmail.com (Gus Gassmann)
Injection-Date: Wed, 17 Aug 2022 16:20:41 +0000
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 by: Gus Gassmann - Wed, 17 Aug 2022 16:20 UTC

On Wednesday, 17 August 2022 at 11:46:34 UTC-3, WM wrote:
> Fritz Feldhase schrieb am Dienstag, 16. August 2022 um 20:16:32 UTC+2:
> > On Tuesday, August 16, 2022 at 5:39:08 PM UTC+2, WM wrote:
> >
> > > The [natural] numbers cannot leave the finite domain.
> >
> > Indeed! And t h e y don't.
> >
> > But the NUMBER of natural numbers can and does.
> The number of natural numbers is a natural number.

You have lost your mind! The number of natural numbers is the cardinality of the set |N, namely aleph_0.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: zelos.ma...@gmail.com (zelos...@gmail.com)
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 by: zelos...@gmail.com - Thu, 18 Aug 2022 04:52 UTC

onsdag 17 augusti 2022 kl. 16:46:34 UTC+2 skrev WM:
> Fritz Feldhase schrieb am Dienstag, 16. August 2022 um 20:16:32 UTC+2:
> > On Tuesday, August 16, 2022 at 5:39:08 PM UTC+2, WM wrote:
> >
> > > The [natural] numbers cannot leave the finite domain.
> >
> > Indeed! And t h e y don't.
> >
> > But the NUMBER of natural numbers can and does.
> The number of natural numbers is a natural number.
> >
> > Hint: There are infinitely many natural numbers.
> Hint: That is potential infinity: Increasing without halt, but always remaining finite.
> >
> > > Proof: {1, 2, 3, ..., n} is finite for every [natrural] number n
> >
> > Indeed!
> >
> > And the set {1, 2, 3, ...} = {n : n e IN} of _all_ natural numbers is _infinite_.
> You need to refrain from individually describing the elements.
> >
> > Moreover: card(1, 2, 3, ...}) = aleph_0 !e IN.
> >
> > You see:
> Only with dark numbers the set can be handled.
>
> Regards, WM
dark numbers are not a thing!

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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 by: WM - Thu, 18 Aug 2022 14:07 UTC

Gus Gassmann schrieb am Mittwoch, 17. August 2022 um 18:20:45 UTC+2:
> On Wednesday, 17 August 2022 at 11:46:34 UTC-3, WM wrote:
> > Fritz Feldhase schrieb am Dienstag, 16. August 2022 um 20:16:32 UTC+2:
> > > On Tuesday, August 16, 2022 at 5:39:08 PM UTC+2, WM wrote:
> > >
> > > > The [natural] numbers cannot leave the finite domain.
> > >
> > > Indeed! And t h e y don't.
> > >
> > > But the NUMBER of natural numbers can and does.
> > The number of natural numbers is a natural number.
> The number of natural numbers is the cardinality of the set |N, namely aleph_0.

The number of definable numbers is less.

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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 by: WM - Thu, 18 Aug 2022 14:08 UTC

William schrieb am Mittwoch, 17. August 2022 um 17:57:29 UTC+2:
> On Wednesday, August 17, 2022 at 11:51:34 AM UTC-3, WM wrote:
>
> > > >The Peano numbers cannot leave the finite domain.
> > > > Proof: {1, 2, 3, ..., n} is finite for every Peano number,
> > > So what? Each *element* of the set of successors is finite. The *set* of successors is infinite
> > and dark.
> Piffle. Each *element* or the set of successors is not "dark" and can be subtracted by a stepwise process.

Then not always infinitely many would remain.

> The fact that the *set* of successors cannnot be subtracted by a stepwise process does not mean there is an *element* of the set of successors that cannot be subtracted by a stepwise process.

There are infinitely many as is proved here
∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo
and here
|ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo.

This does never change. But you cannot define any element that must be remaining. That shows dark numbers.

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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 by: WM - Thu, 18 Aug 2022 14:08 UTC

FromTheRafters schrieb am Mittwoch, 17. August 2022 um 17:32:36 UTC+2:
> WM wrote on 8/17/2022 :
> > Fritz Feldhase schrieb am Dienstag, 16. August 2022 um 20:16:32 UTC+2:
> >> On Tuesday, August 16, 2022 at 5:39:08 PM UTC+2, WM wrote:
> >>
> >>> The [natural] numbers cannot leave the finite domain.
> >>
> >> Indeed! And t h e y don't.
> >>
> >> But the NUMBER of natural numbers can and does.
> >
> > The number of natural numbers is a natural number.
> Good one, tell me another. This time I want unicorns aplenty.

Definable numbers have FISONs. Every FISON has a last number describing its cardinality. It can be proved that there is no actually infinite set

|ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo.

So the number is finite for every FISON.

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: horand.g...@gmail.com (Gus Gassmann)
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 by: Gus Gassmann - Thu, 18 Aug 2022 14:17 UTC

On Thursday, 18 August 2022 at 11:07:31 UTC-3, WM wrote:
[...]
> > The number of natural numbers is the cardinality of the set |N, namely aleph_0.
> The number of definable numbers is less.

The set of definable numbers is decidedly larger than the set of definable _natural_ numbers. If you had a single functioning synspsis left, you would have known that.

Regardless, however, the set of definable natural numbers is a Peano set and hence at least as large as the set of natural numbers. Taken together with the obvious subset relationship, this means that the set of definable natural numbers is in fact equal to the set of natural numbers. This is not changed in the least by your permanent denial.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: wpihug...@gmail.com (William)
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 by: William - Thu, 18 Aug 2022 15:23 UTC

On Thursday, August 18, 2022 at 11:08:13 AM UTC-3, WM wrote:
> William schrieb am Mittwoch, 17. August 2022 um 17:57:29 UTC+2:

> > The fact that the *set* of successors cannnot be subtracted by a stepwise process does not mean there is an *element* of the set of successors that cannot be subtracted by a stepwise process.
> There are infinitely many as is proved here

Piffle. These two statements show that each finite subset of |N is followed by a set of elements with cardinality ℵo (a triviality)

> ∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> and here
> |ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo.
>
> This does never change.

Indeed, the fact that {1,2,3,...,n} ( or {1,2,3,...k,k+1}) is finite never changes. So the two statements do not show that an infinite subset of |N is followed by a set of cardintality ℵo

>But you cannot define any element that must be remaining [after "all elements" the set |N]

There is no such element. |N is not a finite subset of |N

--
William Hughes

Re: Natural numbers and vases III

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From: FTR...@nomail.afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Thu, 18 Aug 2022 11:39:50 -0400
Organization: Peripheral Visions
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 by: FromTheRafters - Thu, 18 Aug 2022 15:39 UTC

WM formulated on Thursday :
> FromTheRafters schrieb am Mittwoch, 17. August 2022 um 17:32:36 UTC+2:
>> WM wrote on 8/17/2022 :
>>> Fritz Feldhase schrieb am Dienstag, 16. August 2022 um 20:16:32 UTC+2:
>>>> On Tuesday, August 16, 2022 at 5:39:08 PM UTC+2, WM wrote:
>>>>
>>>>> The [natural] numbers cannot leave the finite domain.
>>>>
>>>> Indeed! And t h e y don't.
>>>>
>>>> But the NUMBER of natural numbers can and does.
>>>
>>> The number of natural numbers is a natural number.
>> Good one, tell me another. This time I want unicorns aplenty.
>
> Definable numbers have FISONs.

Natural von Neumann constructed ordinal numbers have FISONs. The number
three can be represented as {{{{}}}} also under another construction.

https://en.wikipedia.org/wiki/Natural_number#Zermelo_ordinals

> Every FISON has a last number describing its
> cardinality. It can be proved that there is no actually infinite set

No, such an infinite set cannot be proven to exist, so it is an axiom.
It is hard to disprove an axiom. Good luck with that.

>> ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo.
>
> So the number is finite for every FISON.

So what, they are still associated with countably many of these
endsegments in your sequence of endsegments.

Re: Natural numbers and vases III

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From: FTR...@nomail.afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Thu, 18 Aug 2022 11:42:55 -0400
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 by: FromTheRafters - Thu, 18 Aug 2022 15:42 UTC

WM was thinking very hard :
> Gus Gassmann schrieb am Mittwoch, 17. August 2022 um 18:20:45 UTC+2:
>> On Wednesday, 17 August 2022 at 11:46:34 UTC-3, WM wrote:
>>> Fritz Feldhase schrieb am Dienstag, 16. August 2022 um 20:16:32 UTC+2:
>>>> On Tuesday, August 16, 2022 at 5:39:08 PM UTC+2, WM wrote:
>>>>
>>>>> The [natural] numbers cannot leave the finite domain.
>>>>
>>>> Indeed! And t h e y don't.
>>>>
>>>> But the NUMBER of natural numbers can and does.
>>> The number of natural numbers is a natural number.
>> The number of natural numbers is the cardinality of the set |N, namely
>> aleph_0.
>
> The number of definable numbers is less.

They are already defined, all of them. As such, they were all evidently
definable.

Re: Natural numbers and vases III

<56173f34-206a-45b1-d96c-1350ccb1987f@att.net>

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From: james.g....@att.net (Jim Burns)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Thu, 18 Aug 2022 17:26:59 -0400
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 by: Jim Burns - Thu, 18 Aug 2022 21:26 UTC

On 8/17/2022 10:46 AM, WM wrote:
> Fritz Feldhase schrieb
> am Dienstag, 16. August 2022 um 20:16:32 UTC+2:
>> On Tuesday, August 16, 2022 at 5:39:08 PM UTC+2,
>> WM wrote:

>>> The [natural] numbers cannot leave
>>> the finite domain.
>>
>> Indeed! And t h e y don't.
>> But the NUMBER of natural numbers can
>> and does.
>
> The number of natural numbers is
> a natural number.

A FISON is
a collection with a counting-order
which begins at 0
and which ends _somewhere_

Each FISON 𝐹 has its own FISON-end n
Each FISON-end n has its own FISON 𝐹

Many of the properties of FISON-ends
(AKA natural numbers) arise from
the properties of corresponding FISONs.

For each FISON 𝐹 in 𝓕,
there is a unique FISON-end n in ⋃𝓕
For each FISON-end n in ⋃𝓕,
there is a unique FISON 𝐹 in 𝓕

A FISON ends somewhere.
Each FISON 𝐹 contain some n such that
n+1 is not in 𝐹

However,
𝐹∪{n+1} is also a FISON, meaning also
a collection with a counting-order
which begins at 0
and which ends _somewhere_

𝐹∪{n+1} ∈ 𝓕
n+1 ∈ ⋃𝓕

n+1 ∉ 𝐹
𝐹 ≠ ⋃𝓕

∀𝐹 ∈ 𝓕, 𝐹 ≠ ⋃𝓕
⋃𝓕 ∉ 𝓕

Each FISON is
a collection with a counting-order
which begins at 0
and which ends _somewhere_

The union ⋃𝓕 of FISONs is
a collection with a counting-order
which begins at 0

However,
⋃𝓕 does not end anywhere.
n+1 can be in a different FISON and be in ⋃𝓕.
⋃𝓕 is not a FISON.

The correspondence between FISONs in 𝓕
and FISON-ends in ⋃𝓕 does not apply
to ⋃𝓕 as an element of 𝓕
⋃𝓕 is not an element of ⋃𝓕

The collection of FISON-ends is not a FISON.

> The number of natural numbers is
> a natural number.

The number of natural numbers is NOT
a natural number,
because
the collection of FISON-ends is NOT
a FISON,
because
𝐹∪{n+1} is ALSO a FISON.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: zelos.ma...@gmail.com (zelos...@gmail.com)
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 by: zelos...@gmail.com - Fri, 19 Aug 2022 07:09 UTC

torsdag 18 augusti 2022 kl. 16:07:31 UTC+2 skrev WM:
> Gus Gassmann schrieb am Mittwoch, 17. August 2022 um 18:20:45 UTC+2:
> > On Wednesday, 17 August 2022 at 11:46:34 UTC-3, WM wrote:
> > > Fritz Feldhase schrieb am Dienstag, 16. August 2022 um 20:16:32 UTC+2:
> > > > On Tuesday, August 16, 2022 at 5:39:08 PM UTC+2, WM wrote:
> > > >
> > > > > The [natural] numbers cannot leave the finite domain.
> > > >
> > > > Indeed! And t h e y don't.
> > > >
> > > > But the NUMBER of natural numbers can and does.
> > > The number of natural numbers is a natural number.
> > The number of natural numbers is the cardinality of the set |N, namely aleph_0.
> The number of definable numbers is less.
>
> Regards, WM
your "definable" is meaningless

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: zelos.ma...@gmail.com (zelos...@gmail.com)
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 by: zelos...@gmail.com - Fri, 19 Aug 2022 07:10 UTC

torsdag 18 augusti 2022 kl. 16:08:13 UTC+2 skrev WM:
> William schrieb am Mittwoch, 17. August 2022 um 17:57:29 UTC+2:
> > On Wednesday, August 17, 2022 at 11:51:34 AM UTC-3, WM wrote:
> >
> > > > >The Peano numbers cannot leave the finite domain.
> > > > > Proof: {1, 2, 3, ..., n} is finite for every Peano number,
> > > > So what? Each *element* of the set of successors is finite. The *set* of successors is infinite
> > > and dark.
> > Piffle. Each *element* or the set of successors is not "dark" and can be subtracted by a stepwise process.
> Then not always infinitely many would remain.
> > The fact that the *set* of successors cannnot be subtracted by a stepwise process does not mean there is an *element* of the set of successors that cannot be subtracted by a stepwise process.
> There are infinitely many as is proved here
> ∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> and here
> |ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo.
>
> This does never change. But you cannot define any element that must be remaining. That shows dark numbers.
>
> Regards, WM

No it doesn't! It just shows that cardinal arithmetic works different from ordinary arithmetic

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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 by: WM - Fri, 19 Aug 2022 11:35 UTC

zelos...@gmail.com schrieb am Freitag, 19. August 2022 um 09:10:07 UTC+2:
> torsdag 18 augusti 2022 kl. 16:08:13 UTC+2 skrev WM:
> > William schrieb am Mittwoch, 17. August 2022 um 17:57:29 UTC+2:

> > > The fact that the *set* of successors cannnot be subtracted by a stepwise process does not mean there is an *element* of the set of successors that cannot be subtracted by a stepwise process.
> > There are infinitely many as is proved here
> > ∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> > and here
> > |ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo.
> >
> > This does never change. But you cannot define any element that must be remaining. That shows dark numbers.
> >
> No it doesn't! It just shows that cardinal arithmetic works different from ordinary arithmetic

Logic works the same. This statement
|ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo
says that ℵo numbers are lurking behind all definable numbers. So they are undefinable.

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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 by: WM - Fri, 19 Aug 2022 11:39 UTC

FromTheRafters schrieb am Donnerstag, 18. August 2022 um 17:43:10 UTC+2:
> WM was thinking very hard :
> >> The number of natural numbers is the cardinality of the set |N, namely
> >> aleph_0.
> >
> > The number of definable numbers is less.

> They are already defined, all of them. As such, they were all evidently
> definable.

Not individually. Behind all individually definable numbers there are almost all numbers. They are not individually definable as
∀k ∈ ℕ_deef: |ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo
proves.

Why do you refuse logic?

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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 by: WM - Fri, 19 Aug 2022 11:42 UTC

Gus Gassmann schrieb am Donnerstag, 18. August 2022 um 16:17:58 UTC+2:
> On Thursday, 18 August 2022 at 11:07:31 UTC-3, WM wrote:
> [...]
> > > The number of natural numbers is the cardinality of the set |N, namely aleph_0.
> > The number of definable numbers is less.
> The set of definable numbers is decidedly larger than the set of definable _natural_ numbers.

Not even in set theory. All definable numbers need finite definitions. There are only countably many finite definitions.

Regards, WM

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
Date: Fri, 19 Aug 2022 07:47:23 -0400
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 by: FromTheRafters - Fri, 19 Aug 2022 11:47 UTC

WM formulated on Friday :
> zelos...@gmail.com schrieb am Freitag, 19. August 2022 um 09:10:07 UTC+2:
>> torsdag 18 augusti 2022 kl. 16:08:13 UTC+2 skrev WM:
>>> William schrieb am Mittwoch, 17. August 2022 um 17:57:29 UTC+2:
>
>>>> The fact that the *set* of successors cannnot be subtracted by a stepwise
>>>> process does not mean there is an *element* of the set of successors that
>>>> cannot be subtracted by a stepwise process.
>>> There are infinitely many as is proved here
>>> ∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo
>>> and here
>>>> ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo.
>>>
>>> This does never change. But you cannot define any element that must be
>>> remaining. That shows dark numbers.
>>>
>> No it doesn't! It just shows that cardinal arithmetic works different from
>> ordinary arithmetic
>
> Logic works the same. This statement
>> ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo
> says that ℵo numbers are lurking behind all definable numbers. So they are
> undefinable.

How do you determine if an undefinable thing qualifies as a member of
any particular set?

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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 by: WM - Fri, 19 Aug 2022 11:48 UTC

William schrieb am Donnerstag, 18. August 2022 um 17:23:58 UTC+2:
> On Thursday, August 18, 2022 at 11:08:13 AM UTC-3, WM wrote:
> > William schrieb am Mittwoch, 17. August 2022 um 17:57:29 UTC+2:
>
>
> > > The fact that the *set* of successors cannnot be subtracted by a stepwise process does not mean there is an *element* of the set of successors that cannot be subtracted by a stepwise process.
> > There are infinitely many as is proved here
> Piffle. These two statements show that each finite subset of |N is followed by a set of elements with cardinality ℵo (a triviality)
> > ∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo
> > and here
> > |ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo.
> >
> > This does never change.
> Indeed, the fact that {1,2,3,...,n} ( or {1,2,3,...k,k+1}) is finite never changes. So the two statements do not show that an infinite subset of |N is followed by a set of cardintality ℵo

The set of definable numbers is finite but with not upper threshold. It cannot be ℵo-infinite because
∀k ∈ ℕ: |ℕ \ {1, 2, 3, ..., k}| = ℵo ==> |ℕ \ {1, 2, 3, ..., k, k+1}| = ℵo.

Note: _For all natural numbers_ which are ends of FISONs.
>
> >But you cannot define any element that must be remaining [after "all elements" the set |N]
>
> There is no such element. |N is not a finite subset of |N

Note above: For all derinable natural numbers. (Definable number = end of a FISON.)

Regards, WM

Re: Natural numbers and vases III

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From: FTR...@nomail.afraid.org (FromTheRafters)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Fri, 19 Aug 2022 07:48:07 -0400
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 by: FromTheRafters - Fri, 19 Aug 2022 11:48 UTC

WM wrote :
> FromTheRafters schrieb am Donnerstag, 18. August 2022 um 17:43:10 UTC+2:
>> WM was thinking very hard :
>
>>>> The number of natural numbers is the cardinality of the set |N, namely
>>>> aleph_0.
>>>
>>> The number of definable numbers is less.
>
>> They are already defined, all of them. As such, they were all evidently
>> definable.
>
> Not individually.

who cares?

tech / sci.math / Re: Natural numbers and vases III

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