Re: Two results of set geometry

William Hughes wrote:
On Sep 12, 6:41 am, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> wrote:
WM wrote:
The set of natural numbers is not a natural number. The set N has
omega elements but it has no numerical seize.
The set of natural numbers, if it could exist, would be equivalent to
another natural number. Therefore it cannot exist. (Tip, hint: there
is a bijection between sets and naturals).



Note: this bijection exists in Bruijnland and perhaps
in Wolkensmuekenheim. It does not exist in Orlovia.


- William Hughes


I haven't posted anything for a while. In fact I didn't even touch the group for a while, but recently started reading here and there. It's nice to see Orlovia on the map. I know I'm not forgotten. :)

While it's true that Han's and Wolfgang's complete rejection of the the actually infinite is not one of the areas where I agree with them, I would remind you that I have agreed with Ross's NeN, and Wolfgang's statement that the height of Cantor's list in unary cannot be greater than the width, ie, that there cannot be MORE naturals than the value of any natural. I have repeatedly referred to the naturals as being defined by the equivalence function between element count and value, in close relation to Ross's ER, and the resulting conclusion that the set having no greatest member means it has no definable size. So, that conclusion is valid in the Principality of Orlovia, as well as the County of Hansylvania and the proud Municipality of Meuckenburg. Finlaysonville concurs through mayoral edict. ;)

So, despite the differences between Anti-Cantorians such as Han and WM and Post-Cantorians such as Ross and myself, there is agreement that omega is a phantom created by a faulty logical device based on a difference of 1. The proof of omega using the von Neumann ordinal rests "solidly" on the difference between starting the naturals at 0 or at 1. That shouldn't, and doesn't, matter. The tree always starts with a circle below the root, the gnarl. The circle is always the crux, the foundation thereof. Ask me about that sometime. You can always use my email address, if you so desire. :)

Peace,

Tony Orlow

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