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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
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|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
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|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
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|      |  |   | +- Re: Natural numbers and vases IIIFritz Feldhase
|      |  |   | +- Re: Natural numbers and vases IIIzelos...@gmail.com
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|      |  |   | +- 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
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|      |  |   | +- 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

<tcohd7$tm6$1@gioia.aioe.org>

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From: inva...@invalid.com (Sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Sun, 7 Aug 2022 09:15:03 -0500
Organization: Aioe.org NNTP Server
Message-ID: <tcohd7$tm6$1@gioia.aioe.org>
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 by: Sergio - Sun, 7 Aug 2022 14:15 UTC

On 8/7/2022 7:30 AM, WM wrote:
> William schrieb am Sonntag, 7. August 2022 um 00:33:32 UTC+2:
>> On Saturday, August 6, 2022 at 5:45:47 PM UTC-3, WM wrote:
>>> William schrieb am Samstag, 6. August 2022 um 17:27:50 UTC+2:
>>>
>>>> Let n be a natural number. Writing n down is only one method of showing that n has the property (can be written down)
>>> It is the only reliable way.
>> Piffle.
>
> Even if using induction to define natural numbers you will need dark numbers for the empty intersection. For proof see Natural numbers and vases IV.
>
> Regards, WM

you mean Spoof

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: Sun, 7 Aug 2022 09:16:13 -0500
Organization: Aioe.org NNTP Server
Message-ID: <tcohfd$tm6$2@gioia.aioe.org>
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 by: Sergio - Sun, 7 Aug 2022 14:16 UTC

On 8/7/2022 7:38 AM, WM wrote:
> William schrieb am Sonntag, 7. August 2022 um 00:48:39 UTC+2:
>> On Saturday, August 6, 2022 at 6:01:26 PM UTC-3, WM wrote:
>>> ...Infinite subsets are not subsets of definable endsegments.
>>
>> Piffle. Let S be any subset of endsegments. Each element of S is definable. So S is a "subset of definable endsegments"
>>>> You cannot write down each element of a infinite set, but you can know that each element has the property can be written down (independent of time and resources).
>>> You can have the delusion to know that. But it is wrong.
>> Nope. Induction works for each element of the Infinite set |N.
>
> If it did, then no endsegment would remain after each and all endsegments subject to induction had been omitted.

Nope. Induction works for each element of the Infinite set |N.

>>
>>> ... all definable endsegments leave[s] a set of endsegments having an empty intersection.
>>
>> "all definable endsegments" is a set. Removing this set leaves nothing.
>>
> Induction works for each element, not for the set.

Nope. Induction works for each element of the Infinite set |N.

>
> Regards, WM

Re: Natural numbers and vases III

<tcohh9$tm6$3@gioia.aioe.org>

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From: inva...@invalid.com (Sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Sun, 7 Aug 2022 09:17:13 -0500
Organization: Aioe.org NNTP Server
Message-ID: <tcohh9$tm6$3@gioia.aioe.org>
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 by: Sergio - Sun, 7 Aug 2022 14:17 UTC

On 8/7/2022 7:33 AM, WM wrote:
> FromTheRafters schrieb am Sonntag, 7. August 2022 um 00:40:26 UTC+2:
>> WM wrote on 8/6/2022 :
>
>>> All of them index endsegments which can be omitted from the set of
>>> endsegments. The remainder does not contain any definable endsegment but has
>>> an empty intersection.
>> Because they each have no last element. A last element would be good to
>> indicate an intersection if we were talking about finite sets.
>
> See Natural numbers and vases IV
>
> Regards, WM

for more insulting spoofs...

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: Sun, 07 Aug 2022 11:14:51 -0400
Organization: Peripheral Visions
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 by: FromTheRafters - Sun, 7 Aug 2022 15:14 UTC

WM was thinking very hard :
> FromTheRafters schrieb am Sonntag, 7. August 2022 um 00:40:26 UTC+2:
>> WM wrote on 8/6/2022 :
>
>>> All of them index endsegments which can be omitted from the set of
>>> endsegments. The remainder does not contain any definable endsegment but
>>> has an empty intersection.
>> Because they each have no last element. A last element would be good to
>> indicate an intersection if we were talking about finite sets.
>
> See Natural numbers and vases IV

No!

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 - Sun, 7 Aug 2022 15:37 UTC

On Sunday, August 7, 2022 at 9:38:12 AM UTC-3, WM wrote:
> William schrieb am Sonntag, 7. August 2022 um 00:48:39 UTC+2:
> > On Saturday, August 6, 2022 at 6:01:26 PM UTC-3, WM wrote:
> > >...Infinite subsets are not subsets of definable endsegments.
> >
> > Piffle. Let S be any subset of endsegments. Each element of S is definable. So S is a "subset of definable endsegments"
> > > > You cannot write down each element of a infinite set, but you can know that each element has the property can be written down (independent of time and resources).
> > > You can have the delusion to know that. But it is wrong.
> > Nope. Induction works for each element of the Infinite set |N.
> If it did, then no endsegment would remain after each and all endsegments subject to induction had been [removed].

Correct. Note that "each and all endsegments" is the set of endsegments. When you remove this nothing remains.

--

William Hughes

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 - Sun, 7 Aug 2022 15:45 UTC

On Sunday, August 7, 2022 at 9:38:12 AM UTC-3, WM wrote:
> ∩{E(k) : k > n, k ∈ ℕ, n ∈ ℕ_[def]} = { } .

Correct. Let T be any finite set. Then

> ∩{E(k) : k > n, k ∈ ℕ, n ∈ T} = { } .

Even if we indulge in the fantasy that changable sets exist, we still have that ℕ_def can only change to something that is finite.

--
William Hughes

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 - Sun, 7 Aug 2022 17:54 UTC

On Sunday, August 7, 2022 at 5:45:37 PM UTC+2, William wrote:

> Even if we indulge in the fantasy that changable sets exist, we still have that ℕ_def can only change to something that is finite.

It's called a /variable/ ("restricted" to finite sets).

A /variable/ does not refer to a "changing set".

Re: Natural numbers and vases III

<03241abe-f2f9-492b-81f6-860ebbe40711n@googlegroups.com>

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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 by: WM - Sun, 7 Aug 2022 20:57 UTC

William schrieb am Sonntag, 7. August 2022 um 17:37:54 UTC+2:
> On Sunday, August 7, 2022 at 9:38:12 AM UTC-3, WM wrote:
> > William schrieb am Sonntag, 7. August 2022 um 00:48:39 UTC+2:
> > > On Saturday, August 6, 2022 at 6:01:26 PM UTC-3, WM wrote:
> > > >...Infinite subsets are not subsets of definable endsegments.
> > >
> > > Piffle. Let S be any subset of endsegments. Each element of S is definable. So S is a "subset of definable endsegments"
> > > > > You cannot write down each element of a infinite set, but you can know that each element has the property can be written down (independent of time and resources).
> > > > You can have the delusion to know that. But it is wrong.
> > > Nope. Induction works for each element of the Infinite set |N.
> > If it did, then no endsegment would remain after each and all endsegments subject to induction had been [removed].
>
> Correct. Note that "each and all endsegments" is the set of endsegments. When you remove this nothing remains.
>
How many of its elements can I remove without removing it?

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 - Sun, 7 Aug 2022 20:59 UTC

William schrieb am Sonntag, 7. August 2022 um 17:45:37 UTC+2:
> On Sunday, August 7, 2022 at 9:38:12 AM UTC-3, WM wrote:
> > ∩{E(k) : k > n, k ∈ ℕ, n ∈ ℕ_[def]} = { } .
>
> Correct. Let T be any finite set. Then
>
> > ∩{E(k) : k > n, k ∈ ℕ, n ∈ T} = { } .
>
> Even if we indulge in the fantasy that changable sets exist, we still have that ℕ_def can only change to something that is finite.

Of course. Who could define more individuals!

Regards, WM

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 - Sun, 7 Aug 2022 22:37 UTC

On Sunday, August 7, 2022 at 5:57:36 PM UTC-3, WM wrote:

> How many of ["the set of endsegments"']'s elements can I remove without removing [the set of endsegmentss]?

Who cares? You cannot "remove" the improper subset "each and all endsegments", so there is no contradiction with saying that induction works for each element of the infinite set |N

--
William Hughes

Re: Natural numbers and vases III

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Date: Mon, 8 Aug 2022 06:09:00 -0700 (PDT)
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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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 by: WM - Mon, 8 Aug 2022 13:09 UTC

William schrieb am Montag, 8. August 2022 um 00:37:27 UTC+2:
> On Sunday, August 7, 2022 at 5:57:36 PM UTC-3, WM wrote:
>
> > How many of ["the set of endsegments"']'s elements can I remove without removing [the set of endsegmentss]?
>
> Who cares?

Me.

> You cannot "remove" the improper subset "each and all endsegments", so there is no contradiction with saying that induction works for each element of the infinite set |N

You can remove each element definable by induction. Always more elements remain than you have removed. In order to remove them all, you have to remove the set, i.e., you have to proceed collectively. That is what I say: most element are not removable individually.

Regards, WM

Re: Natural numbers and vases III

<tcr2uj$1t6d$3@gioia.aioe.org>

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https://www.novabbs.com/tech/article-flat.php?id=108490&group=sci.math#108490

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From: inva...@invalid.com (Sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Mon, 8 Aug 2022 08:26:41 -0500
Organization: Aioe.org NNTP Server
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 by: Sergio - Mon, 8 Aug 2022 13:26 UTC

On 8/8/2022 8:09 AM, WM wrote:
> William schrieb am Montag, 8. August 2022 um 00:37:27 UTC+2:
>> On Sunday, August 7, 2022 at 5:57:36 PM UTC-3, WM wrote:
>>
>>> How many of ["the set of endsegments"']'s elements can I remove without removing [the set of endsegmentss]?
>>
>> Who cares?
>
> Me.
>
>> You cannot "remove" the improper subset "each and all endsegments", so there is no contradiction with saying that induction works for each element of the infinite set |N
>
> You can remove each element definable by induction. Always more elements remain than you have removed. In order to remove them all, you have to remove the set, i.e., you have to proceed collectively. That is what I say: most element are not removable individually.
>
> Regards, WM

more trolling, chumming the waters with nonsense

Re: Natural numbers and vases III

<6df90f0b-fc4e-4175-ba6d-648e90f2a2a4n@googlegroups.com>

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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 - Mon, 8 Aug 2022 13:50 UTC

On Monday, 8 August 2022 at 10:09:06 UTC-3, WM wrote:
> William schrieb am Montag, 8. August 2022 um 00:37:27 UTC+2:
> > On Sunday, August 7, 2022 at 5:57:36 PM UTC-3, WM wrote:
> >
> > > How many of ["the set of endsegments"']'s elements can I remove without removing [the set of endsegmentss]?
> >
> > Who cares?
> Me.
> > You cannot "remove" the improper subset "each and all endsegments", so there is no contradiction with saying that induction works for each element of the infinite set |N
> You can remove each element definable by induction. Always more elements remain than you have removed. In order to remove them all, you have to remove the set, i.e., you have to proceed collectively. That is what I say: most element are not removable individually.

Good. An infinite stepwise process cannot be completed one step at a time. Congratulations. How *did* you come to this epiphany?

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 - Mon, 8 Aug 2022 15:46 UTC

On Monday, August 8, 2022 at 10:09:06 AM UTC-3, WM wrote:
> William schrieb am Montag, 8. August 2022 um 00:37:27 UTC+2:
> > On Sunday, August 7, 2022 at 5:57:36 PM UTC-3, WM wrote:
> >

> > You cannot "remove" the improper subset "each and all endsegments", so there is no contradiction with saying that induction works for each element of the infinite set |N
> You can remove each element definable by induction. Always more elements remain than you have removed. In order to remove them all, you have to remove the set.

If you "remove them all" nothing remains, so the intersection is empty. There is no contradiction. Each element of |N is subject to induction.

--
William Hughes

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 - Mon, 8 Aug 2022 17:44 UTC

On Monday, August 8, 2022 at 5:46:56 PM UTC+2, William wrote:

> If you "remove them all" nothing remains, so the intersection is

not defined (in ZFC). Of course, still no contradiction.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
Injection-Date: Tue, 09 Aug 2022 10:51:23 +0000
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 by: WM - Tue, 9 Aug 2022 10:51 UTC

Gus Gassmann schrieb am Montag, 8. August 2022 um 15:50:30 UTC+2:
> On Monday, 8 August 2022 at 10:09:06 UTC-3, WM wrote:
> > William schrieb am Montag, 8. August 2022 um 00:37:27 UTC+2:
> > > On Sunday, August 7, 2022 at 5:57:36 PM UTC-3, WM wrote:
> > >
> > > > How many of ["the set of endsegments"']'s elements can I remove without removing [the set of endsegmentss]?
> > >
> > > Who cares?
> > Me.
> > > You cannot "remove" the improper subset "each and all endsegments", so there is no contradiction with saying that induction works for each element of the infinite set |N
> > You can remove each element definable by induction. Always more elements remain than you have removed. In order to remove them all, you have to remove the set, i.e., you have to proceed collectively. That is what I say: most element are not removable individually.
> Good. An infinite stepwise process cannot be completed one step at a time..

One step at every half time should be possible. But it is not.

XOOOO...
XOOOO...
XOOOO...
XOOOO...
XOOOO...
....

will never become

XXX...
XXX...
XXX...
....

by exchanges of X and O.

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 - Tue, 9 Aug 2022 10:55 UTC

William schrieb am Montag, 8. August 2022 um 17:46:56 UTC+2:
> On Monday, August 8, 2022 at 10:09:06 AM UTC-3, WM wrote:
> > William schrieb am Montag, 8. August 2022 um 00:37:27 UTC+2:
> > > On Sunday, August 7, 2022 at 5:57:36 PM UTC-3, WM wrote:
> > >
>
> > > You cannot "remove" the improper subset "each and all endsegments", so there is no contradiction with saying that induction works for each element of the infinite set |N
> > You can remove each element definable by induction. Always more elements remain than you have removed. In order to remove them all, you have to remove the set.
>
> If you "remove them all" nothing remains, so the intersection is empty.

Of course.

> There is no contradiction. Each element of |N is subject to induction.

No. Induction proves that never an empty intersection will be gained. For all inductively accessible endsegments we prove

∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀ .

Hence

|∩{E(k) : k ∈ ℕ_def}| = ℵ₀ .

Only for the infinite, i.e., not individually definable set of endsegments it is assumed that

∩{E(k) : k ∈ ℕ} = { } .

Regards, WM

Re: Natural numbers and vases III

<48c3a08a-85d5-4c2f-98b6-c35c4bbc0d7an@googlegroups.com>

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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 by: WM - Tue, 9 Aug 2022 10:56 UTC

Fritz Feldhase schrieb am Montag, 8. August 2022 um 19:44:46 UTC+2:
> On Monday, August 8, 2022 at 5:46:56 PM UTC+2, William wrote:
>
> > If you "remove them all" nothing remains, so the intersection is
> not defined (in ZFC). Of course, still no contradiction.

But this is a contracition: Induction proves that never an empty intersection will be gained. For all inductively accessible endsegments we prove

∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀ .

Hence

|∩{E(k) : k ∈ ℕ_def}| = ℵ₀ .

Only for the infinite, i.e., not individually definable set of endsegments it is assumed that

∩{E(k) : k ∈ ℕ} = { } .

Regards, WM

Re: Natural numbers and vases III

<1a7f8a63-39d4-404c-84aa-32765b86d42bn@googlegroups.com>

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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 - Tue, 9 Aug 2022 11:14 UTC

tisdag 9 augusti 2022 kl. 12:56:29 UTC+2 skrev WM:
> Fritz Feldhase schrieb am Montag, 8. August 2022 um 19:44:46 UTC+2:
> > On Monday, August 8, 2022 at 5:46:56 PM UTC+2, William wrote:
> >
> > > If you "remove them all" nothing remains, so the intersection is
> > not defined (in ZFC). Of course, still no contradiction.
> But this is a contracition: Induction proves that never an empty intersection will be gained. For all inductively accessible endsegments we prove
> ∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀ .
>
> Hence
>
> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀ .
>
> Only for the infinite, i.e., not individually definable set of endsegments it is assumed that
>
> ∩{E(k) : k ∈ ℕ} = { } .
>
> Regards, WM
it proves only that for any FINITE intersection it is non-empty, not that the INFINITE intersection is non-empty.

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 - Tue, 9 Aug 2022 11:54 UTC

zelos...@gmail.com schrieb am Dienstag, 9. August 2022 um 13:14:16 UTC+2:
> tisdag 9 augusti 2022 kl. 12:56:29 UTC+2 skrev WM:
> > Fritz Feldhase schrieb am Montag, 8. August 2022 um 19:44:46 UTC+2:
> > > On Monday, August 8, 2022 at 5:46:56 PM UTC+2, William wrote:
> > >
> > > > If you "remove them all" nothing remains, so the intersection is
> > > not defined (in ZFC). Of course, still no contradiction.
> > But this is a contracition: Induction proves that never an empty intersection will be gained. For all inductively accessible endsegments we prove
> > ∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀ .
> >
> > Hence
> >
> > |∩{E(k) : k ∈ ℕ_def}| = ℵ₀ .
> >
> > Only for the infinite, i.e., not individually definable set of endsegments it is assumed that
> >
> > ∩{E(k) : k ∈ ℕ} = { } .

> it proves only that for any FINITE intersection it is non-empty, not that the INFINITE intersection is non-empty.

Of course. All intersections with definable endsegments are finite. The infinite intersection contains endsegments which are not individually definable.

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 - Tue, 9 Aug 2022 12:15 UTC

On Tuesday, 9 August 2022 at 07:51:27 UTC-3, WM wrote:
> XOOOO...
> XOOOO...
> XOOOO...
> XOOOO...
> XOOOO...
> ...
>
> will never become
>
> XXX...
> XXX...
> XXX...
> ...
>
> by exchanges of X and O.

Because you say so? Bullshit! Once an X appears anywhere to the right of the first column, there is no need to move it again, and the O's are gradually pushed further and further down the main column at the same time.

If you slot the X's from the first column into the rest of the matrix in the order given by

1 2 4 7 ...
3 5 8 ...
6 9 ...
10 ...
....

the limit matrix will have all X's, and the O's will all have disappeared. Too hard for you , or just inconvenient?

Re: Natural numbers and vases III

<7f97b550-e649-4c4f-943c-ea70ed99ea44n@googlegroups.com>

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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, 9 Aug 2022 13:13 UTC

On Tuesday, August 9, 2022 at 2:15:21 PM UTC+2, Gus Gassmann wrote:
> On Tuesday, 9 August 2022 at 07:51:27 UTC-3, WM wrote:
> >
> > XOOOO...
> > XOOOO...
> > XOOOO...
> > XOOOO...
> > XOOOO...
> > ...
> >
> > will never become
> >
> > XXX...
> > XXX...
> > XXX...
> > ...
> >
> > by exchanges of X and O.
> >
> Because you say so?

Because he is right.

Hint: If there is an O in the matrix A then there's an O in the matrix swap(i,j;k,l)(A).

Re: Natural numbers and vases III

<tctnle$6c4$1@gioia.aioe.org>

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https://www.novabbs.com/tech/article-flat.php?id=108634&group=sci.math#108634

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From: inva...@invalid.com (Sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Tue, 9 Aug 2022 08:32:29 -0500
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 by: Sergio - Tue, 9 Aug 2022 13:32 UTC

On 8/9/2022 5:55 AM, WM wrote:
> William schrieb am Montag, 8. August 2022 um 17:46:56 UTC+2:
>> On Monday, August 8, 2022 at 10:09:06 AM UTC-3, WM wrote:
>>> William schrieb am Montag, 8. August 2022 um 00:37:27 UTC+2:
>>>> On Sunday, August 7, 2022 at 5:57:36 PM UTC-3, WM wrote:
>>>>
>>
>>>> You cannot "remove" the improper subset "each and all endsegments", so there is no contradiction with saying that induction works for each element of the infinite set |N
>>> You can remove each element definable by induction. Always more elements remain than you have removed. In order to remove them all, you have to remove the set.
>>
>> If you "remove them all" nothing remains, so the intersection is empty.
>
> Of course.
>
>> There is no contradiction. Each element of |N is subject to induction.
>
> No. Induction proves that never an empty intersection will be gained. For all inductively accessible endsegments we prove
>
> ∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀ .

ok, but that is not induction.

>
> Hence
>
> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀ .

wrong, intersection of all endsegments is empty.

your insistence on bad math underscores your rotten attitude towards Math, you must be fired.

>

> Regards, WM
>

Re: Natural numbers and vases III

<ca10f365-358f-4dc9-ad29-2b154b0cfb9bn@googlegroups.com>

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Subject: Re: Natural numbers and vases III
From: horand.g...@gmail.com (Gus Gassmann)
Injection-Date: Tue, 09 Aug 2022 13:32:47 +0000
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 by: Gus Gassmann - Tue, 9 Aug 2022 13:32 UTC

On Tuesday, 9 August 2022 at 10:13:13 UTC-3, Fritz Feldhase wrote:
> On Tuesday, August 9, 2022 at 2:15:21 PM UTC+2, Gus Gassmann wrote:
> > On Tuesday, 9 August 2022 at 07:51:27 UTC-3, WM wrote:
> > >
> > > XOOOO...
> > > XOOOO...
> > > XOOOO...
> > > XOOOO...
> > > XOOOO...
> > > ...
> > >
> > > will never become
> > >
> > > XXX...
> > > XXX...
> > > XXX...
> > > ...
> > >
> > > by exchanges of X and O.
> > >
> > Because you say so?
> Because he is right.
>
> Hint: If there is an O in the matrix A then there's an O in the matrix swap(i,j;k,l)(A).

I talked about the *limit*. Why did you snip that?

All you have to do is define the convergence criterion properly, for instance using the distance function d(a,b) = 0 if a = b, = 1 otherwise (for a,b in {O, X}).

For every i,j there exists a K_ij such that M(k)_{ij} = X for k > K_ij. That is pointwise convergence to

XXX...
XXX...
XXX...
....

Re: Natural numbers and vases III

<tctnnj$6c4$2@gioia.aioe.org>

  copy mid

https://www.novabbs.com/tech/article-flat.php?id=108636&group=sci.math#108636

  copy link   Newsgroups: sci.math
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From: inva...@invalid.com (Sergio)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Tue, 9 Aug 2022 08:33:38 -0500
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 by: Sergio - Tue, 9 Aug 2022 13:33 UTC

On 8/9/2022 6:54 AM, WM wrote:
> zelos...@gmail.com schrieb am Dienstag, 9. August 2022 um 13:14:16 UTC+2:
>> tisdag 9 augusti 2022 kl. 12:56:29 UTC+2 skrev WM:
>>> Fritz Feldhase schrieb am Montag, 8. August 2022 um 19:44:46 UTC+2:
>>>> On Monday, August 8, 2022 at 5:46:56 PM UTC+2, William wrote:
>>>>
>>>>> If you "remove them all" nothing remains, so the intersection is
>>>> not defined (in ZFC). Of course, still no contradiction.
>>> But this is a contracition: Induction proves that never an empty intersection will be gained. For all inductively accessible endsegments we prove
>>> ∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀ .
>>>
>>> Hence
>>>
>>> |∩{E(k) : k ∈ ℕ_def}| = ℵ₀ .
>>>
>>> Only for the infinite, i.e., not individually definable set of endsegments it is assumed that
>>>
>>> ∩{E(k) : k ∈ ℕ} = { } .
>
>> it proves only that for any FINITE intersection it is non-empty, not that the INFINITE intersection is non-empty.
>

<snip nonsense>

>
> Regards, WM

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

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