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A complex system that works is invariably found to have evolved from a simple system that works.

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

<103fa899-4678-453c-94e4-011d89c3e94an@googlegroups.com>

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Subject: Re: Natural numbers and vases III
From: wpihug...@gmail.com (William)
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 by: William - Tue, 19 Jul 2022 15:39 UTC

On Tuesday, July 19, 2022 at 12:09:01 PM UTC-3, WM wrote:
> William schrieb am Dienstag, 19. Juli 2022 um 16:37:41 UTC+2:
> > On Tuesday, July 19, 2022 at 9:16:10 AM UTC-3, WM wrote:
> >
> >
> > > Note that "three elements of the set" is not an element of the set. Nevertheless three elements can be removed.
> > Correct. If X is not an element of the set it may or may not be able to be removed , e.g. if X is a subset of |N it can be removed if X is finite.
> Who made up this restriction?

It is not a restriction. Some infinite subsets can be subtracted and an infinite set is left, others, e.g. "all elements of |N" cannot.

<snip>

> Defined = finite,

By trivial induction each element of "all elements of |N" (without exception) is finite.

--
William Hughes

Re: Natural numbers and vases III

<01c579e1-60ed-499c-8294-069f117dc653n@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, 19 Jul 2022 15:40 UTC

William schrieb am Dienstag, 19. Juli 2022 um 17:39:18 UTC+2:
> On Tuesday, July 19, 2022 at 12:09:01 PM UTC-3, WM wrote:
> > William schrieb am Dienstag, 19. Juli 2022 um 16:37:41 UTC+2:
> > > On Tuesday, July 19, 2022 at 9:16:10 AM UTC-3, WM wrote:
> > >
> > >
> > > > Note that "three elements of the set" is not an element of the set. Nevertheless three elements can be removed.
> > > Correct. If X is not an element of the set it may or may not be able to be removed , e.g. if X is a subset of |N it can be removed if X is finite.
> > Who made up this restriction?
> It is not a restriction. Some infinite subsets can be subtracted and an infinite set is left, others, e.g. "all elements of |N" cannot.

Why?
>
> <snip>
>
> > Defined = finite,
>
> By trivial induction each element of "all elements of |N" (without exception) is finite.

Subset of defined elements = finite subset.

Regards, WM

Re: Natural numbers and vases III

<ef2efaf0-050e-4801-83b0-92a37dedf903n@googlegroups.com>

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Subject: Re: Natural numbers and vases III
From: wpihug...@gmail.com (William)
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 by: William - Tue, 19 Jul 2022 15:53 UTC

On Tuesday, July 19, 2022 at 12:40:52 PM UTC-3, WM wrote:
> William schrieb am Dienstag, 19. Juli 2022 um 17:39:18 UTC+2:
> > On Tuesday, July 19, 2022 at 12:09:01 PM UTC-3, WM wrote:
> > > William schrieb am Dienstag, 19. Juli 2022 um 16:37:41 UTC+2:
> > > > On Tuesday, July 19, 2022 at 9:16:10 AM UTC-3, WM wrote:
> > > >
> > > >
> > > > > Note that "three elements of the set" is not an element of the set. Nevertheless three elements can be removed.
> > > > Correct. If X is not an element of the set it may or may not be able to be removed , e.g. if X is a subset of |N it can be removed if X is finite.
> > > Who made up this restriction?
> > It is not a restriction. Some infinite subsets can be subtracted and an infinite set is left, others, e.g. "all elements of |N" cannot.
> Why?

"all elements of |N" is the set |N, so:
|N\"all elements of |N"=|N\|N={}.

<snip>

> Subset of defined elements = finite subset.

False. Each element of "all elements of |N" is finite (trivial induction). Thus "all elements of |N" is a subset of defined elements but "all elements of |N" is not a finite subset.
--
William Hughes

Re: Natural numbers and vases III

<4ed25630-bafc-445f-8ecb-e54a3f0841aen@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, 19 Jul 2022 16:06 UTC

William schrieb am Dienstag, 19. Juli 2022 um 17:53:18 UTC+2:
> On Tuesday, July 19, 2022 at 12:40:52 PM UTC-3, WM wrote:
> > William schrieb am Dienstag, 19. Juli 2022 um 17:39:18 UTC+2:
> > > On Tuesday, July 19, 2022 at 12:09:01 PM UTC-3, WM wrote:
> > > > William schrieb am Dienstag, 19. Juli 2022 um 16:37:41 UTC+2:
> > > > > On Tuesday, July 19, 2022 at 9:16:10 AM UTC-3, WM wrote:
> > > > >
> > > > >
> > > > > > Note that "three elements of the set" is not an element of the set. Nevertheless three elements can be removed.
> > > > > Correct. If X is not an element of the set it may or may not be able to be removed , e.g. if X is a subset of |N it can be removed if X is finite.
> > > > Who made up this restriction?
> > > It is not a restriction. Some infinite subsets can be subtracted and an infinite set is left, others, e.g. "all elements of |N" cannot.
> > Why?
> "all elements of |N" is the set |N, so:
> |N\"all elements of |N"=|N\|N={}.
>
> <snip>
> > Subset of defined elements = finite subset.
> False. Each element of "all elements of |N" is finite (trivial induction).

But not each is definable.

> Thus "all elements of |N" is a subset of defined elements

Nonsense. Define more than finitely many individuals. If you could, you could remove all elements of |N without removing |N. But you can't.

> but "all elements of |N" is not a finite subset.

Subsets of definable elements are finite. Proof by failure of defining more.

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 - Tue, 19 Jul 2022 16:30 UTC

On Tuesday, July 19, 2022 at 1:06:09 PM UTC-3, WM wrote:
> William schrieb am Dienstag, 19. Juli 2022 um 17:53:18 UTC+2:
> > On Tuesday, July 19, 2022 at 12:40:52 PM UTC-3, WM wrote:
> > > William schrieb am Dienstag, 19. Juli 2022 um 17:39:18 UTC+2:
> > > > On Tuesday, July 19, 2022 at 12:09:01 PM UTC-3, WM wrote:
> > > > > William schrieb am Dienstag, 19. Juli 2022 um 16:37:41 UTC+2:
> > > > > > On Tuesday, July 19, 2022 at 9:16:10 AM UTC-3, WM wrote:
> > > > > >
> > > > > >
> > > > > > > Note that "three elements of the set" is not an element of the set. Nevertheless three elements can be removed.
> > > > > > Correct. If X is not an element of the set it may or may not be able to be removed , e.g. if X is a subset of |N it can be removed if X is finite.
> > > > > Who made up this restriction?
> > > > It is not a restriction. Some infinite subsets can be subtracted and an infinite set is left, others, e.g. "all elements of |N" cannot.
> > > Why?
> > "all elements of |N" is the set |N, so:
> > |N\"all elements of |N"=|N\|N={}.
> >
> > <snip>
> > > Subset of defined elements = finite subset.
> > False. Each element of "all elements of |N" is finite (trivial induction).
> But not each is definable.

Piffle,

WM: Defined = 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 - Tue, 19 Jul 2022 18:02 UTC

On Tuesday, July 19, 2022 at 5:53:18 PM UTC+2, William wrote:

> "all elements of |N" is the set |N, so:

Actually, no.

THE SET OF all elemens of IN is the set IN.

{n : n e IN} = IN.

But 1, 2, 3, ... =/= IN.

Hint: All elements of IN are elements of Z. But IN is not an element of Z.

Try to get that straight, William.

Re: Natural numbers and vases III

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Subject: Re: Natural numbers and vases III
Date: Tue, 19 Jul 2022 13:47:10 -0500
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 by: sergi o - Tue, 19 Jul 2022 18:47 UTC

On 7/19/2022 1:02 PM, Fritz Feldhase wrote:
> On Tuesday, July 19, 2022 at 5:53:18 PM UTC+2, William wrote:
>
>> "all elements of |N" is the set |N, so:
>
> Actually, no.
>
> THE SET OF all elemens of IN is the set IN.
>
> {n : n e IN} = IN.
>
> But 1, 2, 3, ... =/= IN.
>
> Hint: All elements of IN are elements of Z. But IN is not an element of Z.
>
> Try to get that straight, William.

1,2,3,4,...

or {1,2,3,4,...}

or {1},{2},{3},{4},...

math nomenclature.

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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 by: WM - Tue, 19 Jul 2022 18:47 UTC

Fritz Feldhase schrieb am Dienstag, 19. Juli 2022 um 20:02:49 UTC+2:
> On Tuesday, July 19, 2022 at 5:53:18 PM UTC+2, William wrote:
>
> > "all elements of |N" is the set |N, so:
> Actually, no.

What else is in |N?

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, 19 Jul 2022 18:49 UTC

William schrieb am Dienstag, 19. Juli 2022 um 18:30:29 UTC+2:

> WM: Defined = finite,

as an abbreviation of: subset of defined elements = finite subset.

Regards, WM

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, 19 Jul 2022 19:19 UTC

On Tuesday, July 19, 2022 at 8:47:37 PM UTC+2, sergi o wrote:
> On 7/19/2022 1:02 PM, Fritz Feldhase wrote:
> > On Tuesday, July 19, 2022 at 5:53:18 PM UTC+2, William wrote:
> >
> >> "all elements of |N" is the set |N, so:
> >
> > Actually, no.
> >
> > THE SET OF all elemens of IN is the set IN.
> >
> > {n : n e IN} = IN.
> >
> > But 1, 2, 3, ... =/= IN.

Actually, even that statement is nonsensical already. :-)

> > Hint: All elements of IN are elements of Z. But IN is not an element of Z.
> >
> > Try to get that straight, William.
> >
> 1,2,3,4,...

Clearly referring to all natural numbers, but not to the SET of all natural numbers.

Hint: The latter can be denoted with "{1, 2, 3, 4, ....}"

You see, this does NOT mean IN = {IN}.
> or {1, 2, 3, 4,...}

Usually denoting the set of all natural numbers, i. e. IN.

> or {1}, {2}, {3}, {4},...

referring to the/all singletons with a natural number as sole element.

But not referring to IN, nor to the set {{1}, {2}, {3},{ 4},...} = {{n} : n e IN}.

> math nomenclature

Indeed! It's important to use CORRECT language and notation in math.

ESPECIALLY when "discussing" with cranks.

Hint: "Cranks who contradict some mainstream opinion in some highly technical field, (e.g. mathematics) frequently [...] misunderstand or do not use standard notation and terminology."

At least WE should do better.

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 - Tue, 19 Jul 2022 19:29 UTC

We can use induction to show that:

1. Each element (without exception) is in the bijection.
2. Each element (without exception) is followed by an infinite number of successors.

However

"All elements of |N" ( i.e. the set |N) is not followed by an element of |N.

--
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 - Tue, 19 Jul 2022 20:14 UTC

> "All elements of |N" ( i.e. the set |N)

Nonsense. Holy shit!

You guys are going nuts. To much Mückenheim, it seems.

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, 19 Jul 2022 20:15 UTC

> "All elements of |N" ( i.e. the set |N)

Nonsense. Holy shit!

You guys are going nuts. Too much Mückenheim, it seems.

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, 19 Jul 2022 20:23 UTC

On Tuesday, July 19, 2022 at 9:30:04 PM UTC+2, William wrote:

> 1. Each element (without exception) is in the bijection.

"is in the bijection"? HOLY SHIT!

Like "is in your asshole", or what?

Ever learned to use math lingo?

Hint: Something might be in the DOMAIN of a bijection, or in its IMAGE or in its CODOMAIN.

So what do you MEAN by that IDIOTIC claim?

_______________________________

From a set-theoretic point of view the pairs (x, f(x)) with x e dom(f) are IN the bijection f (if f is a bijection).

But I guess that's NOT what you mean with 1. So WHAT THE HELL _do_ you mean with this mumbo-jumbo?

Why do you talk like a crank/Mückenheim?

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, 20 Jul 2022 04:55 UTC

tisdag 19 juli 2022 kl. 14:21:25 UTC+2 skrev WM:
> zelos...@gmail.com schrieb am Dienstag, 19. Juli 2022 um 04:47:04 UTC+2:
> > måndag 18 juli 2022 kl. 23:29:37 UTC+2 skrev WM:
> > > William schrieb am Montag, 18. Juli 2022 um 20:17:17 UTC+2:
> > > > On Monday, July 18, 2022 at 2:47:34 PM UTC-3, WM wrote:
> > >
> > > > Let |N_b be the subset of |N of elements that are in the bijection,
> > > > Let |N_s be the subset of |N of elements that have a successor
> > > > > > Simple induction shows that 1. each element (without exception) is in the bijection
> > >
> > > By the way, this is wrong. Simple induction shows that every element in the bijection has infinitely many successors.
> > They do, but they are still equal to N
> Of course. Dark successors and definable elements are the element of |N.

There is nothing "dark"

> > > We don't remove the set but all its elements. If this is not the same, then something is wrong in Cantor-land.
> Regards, WM

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, 20 Jul 2022 04:55 UTC

tisdag 19 juli 2022 kl. 14:27:34 UTC+2 skrev WM:
> Eram semper recta schrieb am Dienstag, 19. Juli 2022 um 05:06:47 UTC+2:
>
> > Set theory never actually succeeded.
> Unfortunately set theory has actually succeeded to fill millions with enthusiasm (including myself) and to distract them from serious work.
>
> Regards, WM
no, it WORKS, period. You being too stupid to understand it does not invalidate it

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From: zelos.ma...@gmail.com (zelos...@gmail.com)
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 by: zelos...@gmail.com - Wed, 20 Jul 2022 04:56 UTC

tisdag 19 juli 2022 kl. 20:49:41 UTC+2 skrev WM:
> William schrieb am Dienstag, 19. Juli 2022 um 18:30:29 UTC+2:
>
> > WM: Defined = finite,
>
> as an abbreviation of: subset of defined elements = finite subset.
>
> Regards, WM
which means it has a largest element.

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 - Wed, 20 Jul 2022 11:14 UTC

zelos...@gmail.com schrieb am Mittwoch, 20. Juli 2022 um 06:56:48 UTC+2:
> tisdag 19 juli 2022 kl. 20:49:41 UTC+2 skrev WM:
> > William schrieb am Dienstag, 19. Juli 2022 um 18:30:29 UTC+2:
> >
> > > WM: Defined = finite,
> >
> > as an abbreviation of: subset of defined elements = finite subset.
> >
> which means it has a largest element.

Try to find it. I will give you a larger one.

Regards, WM

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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 by: WM - Wed, 20 Jul 2022 11:22 UTC

Fritz Feldhase schrieb am Dienstag, 19. Juli 2022 um 22:23:48 UTC+2:

> Hint: Something might be in the DOMAIN of a bijection, or in its IMAGE or in its CODOMAIN.
Then it is in the bijection too.
5 ∈ ℕ ⊂ ℝ ⊂ mathematical objects ==> 5 is in ℝ because it is a real number and 5 is in the mathematical objects with no doubt.

Regards, WM

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Subject: Re: Natural numbers and vases III
From: askaske...@gmail.com (WM)
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 by: WM - Wed, 20 Jul 2022 11:25 UTC

zelos...@gmail.com schrieb am Mittwoch, 20. Juli 2022 um 06:55:20 UTC+2:
> tisdag 19 juli 2022 kl. 14:21:25 UTC+2 skrev WM:
> > Dark successors and definable elements are the elements of |N.
> There is nothing "dark"

Proof: ∩{E(k) : k ∈ ℕ} = { }
But apply the Minorantenkriterium for infinite sequences.
There is a sequence (S(k)) where S(k) = |∩{E(1), E(2), ..., E(k)} = ℵ₀ .
This sequence has a minorante
(M(k)) where M(k) = 10 for all k.

According to mathematics, the limit of (S(k)) cannot be less than the limit of (M(k)), if (S(k)) has a limit at all.

> > > > We don't remove the set but all its elements. If this is not the same, then something is wrong in Cantor-land.

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 - Wed, 20 Jul 2022 11:29 UTC

William schrieb am Dienstag, 19. Juli 2022 um 21:30:04 UTC+2:
> We can use induction to show that:
>
> 1. Each element (without exception) is in the bijection.

If you remove all definable natural numbers, then infinitely many remain.
> 2. Each element (without exception) is followed by an infinite number of successors.
>
> However
>
> "All elements of |N" ( i.e. the set |N) is not followed by an element of |N.

If removing all elements differs from removing the set, then something is rotten to the core in Cantor-land. By what should it differ?

Regards, WM

Re: Natural numbers and vases III

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 by: sergi o - Wed, 20 Jul 2022 12:58 UTC

On 7/20/2022 6:22 AM, WM wrote:
> Fritz Feldhase schrieb am Dienstag, 19. Juli 2022 um 22:23:48 UTC+2:
>
>> Hint: Something might be in the DOMAIN of a bijection, or in its IMAGE or in its CODOMAIN.
> Then it is in the bijection too.
> 5 ∈ ℕ ⊂ ℝ ⊂ mathematical objects ==> 5 is in ℝ because it is a real number and 5 is in the mathematical objects with no doubt.

IAW WM, how can 5 be a real number ? as WM will argue that 5 is not specified without all its digits, 5.0000000000000....

AND preceeding digits, ...00000000000005.00000000000...

Therefore, IAW WM, we will wait until you have written down all the digits first, before using the number.

>
> Regards, WM

IAW => In Accordance With

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 by: sergi o - Wed, 20 Jul 2022 13:01 UTC

On 7/20/2022 6:29 AM, WM wrote:
> William schrieb am Dienstag, 19. Juli 2022 um 21:30:04 UTC+2:
>> We can use induction to show that:
>>
>> 1. Each element (without exception) is in the bijection.
>
> If you remove all definable-QUAcK!! natural numbers, then infinitely many remain.

False statement.

>
>> 2. Each element (without exception) is followed by an infinite number of successors.
>>
>> However
>>
>> "All elements of |N" ( i.e. the set |N) is not followed by an element of |N.

No, any element of |N" ( i.e. the set |N) is followed by an element of |N.

>
> If removing all elements differs from removing the set, then something is rotten to the core in Cantor-land.

No. Its rotten to the core in WM-land

>
> Regards, WM

Re: Natural numbers and vases III

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 by: sergi o - Wed, 20 Jul 2022 13:02 UTC

On 7/20/2022 6:14 AM, WM wrote:
> zelos...@gmail.com schrieb am Mittwoch, 20. Juli 2022 um 06:56:48 UTC+2:
>> tisdag 19 juli 2022 kl. 20:49:41 UTC+2 skrev WM:
>>> William schrieb am Dienstag, 19. Juli 2022 um 18:30:29 UTC+2:
>>>
>>>> WM: Defined = finite,
>>>
>>> as an abbreviation of: subset of defined elements = finite subset.
>>>
>> which means it has a largest element.
>
> Try to find it. I will give you a larger one.

which is a larger finite set.

>
> Regards, WM

Re: Natural numbers and vases III

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From: inva...@invalid.com (sergi o)
Newsgroups: sci.math
Subject: Re: Natural numbers and vases III
Date: Wed, 20 Jul 2022 08:22:30 -0500
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 by: sergi o - Wed, 20 Jul 2022 13:22 UTC

On 7/20/2022 6:25 AM, WM wrote:
> zelos...@gmail.com schrieb am Mittwoch, 20. Juli 2022 um 06:55:20 UTC+2:
>> tisdag 19 juli 2022 kl. 14:21:25 UTC+2 skrev WM:
>>> Dark successors and definable elements are the elements of |N.
>> There is nothing "dark"
>
> Proof: ∩{E(k) : k ∈ ℕ} = { }
> But apply the Minorantenkriterium for infinite sequences.
> There is a sequence (S(k)) where S(k) = |∩{E(1), E(2), ..., E(k)} = ℵ₀ .

the equation is written wrong, please correct it, sloppy math.

> This sequence has a *minorante*

spelling error, dont you mean lower bound >

> (M(k)) where M(k) = 10 for all k.

where did 10 come from ? descended from the heavens ?

>
> According to mathematics,

(that means here comes more BS)

> the limit of (S(k)) cannot be less than the limit of (M(k)), if (S(k)) has a limit at all.

so M(k) = 10 for all k, why use M(k) at all ?? just use 10. smoke and mirrors?

So now you are at S(k) > 10

now what ? anything ? stuck ? need some hints ?

******** => HINT #1 <= ***********

you arbitrarily assigned a lower bound to a sequence that is infinite, and got stuck

******** => HINT #2 <= ***********

memorize these;

S(k) = |∩{E(1), E(2), ..., E(k)}| = ℵ₀
S(k) = |∩{E(1), E(2), ...,E(k),...}| = {}

******** => HINT #3 <= ***********

M(k) is poopus

**Extra Credit Problem:

What is the difference between these two equations ?

S(k) = |∩{E(1), E(2), ..., E(k)}| = ℵ₀
S(k) = |∩{E(1), E(2), ...,E(k),...}| = {}

no more spoon feeding today for you!

> Regards, WM

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

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