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Topic: Few questions on forcing, large cardinals
Replies: 17   Last Post: Mar 30, 2013 1:21 PM

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fom

Posts: 1,968
Registered: 12/4/12
Re: Few questions on forcing, large cardinals
Posted: Mar 23, 2013 6:34 PM
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On 3/23/2013 5:09 PM, Ross A. Finlayson wrote:
> On Mar 23, 2:44 pm, fom <fomJ...@nyms.net> wrote:
>> On 3/23/2013 4:34 PM, Ross A. Finlayson wrote:
>>
>>
>>

>>> In a sense, infinity _is_ the numbers. Start from even more
>>> fundamental objects than natural numbers as elements. Like the
>>> numbers, they are as different as they can be and as same as they can
>>> be, where they are each different in not being any other and each same
>>> in being defined by that difference. There's no stop to that, it's
>>> gone on, forever. Then, in a way like when you look into the void, it
>>> looks into you

>>
>> Is this your way of saying that if you look
>> into the void, you and the void become one?
>>
>> http://en.wikipedia.org/wiki/Cantor-Bernstein-Schroeder_theorem
>>
>> Just kiddding....
>>
>> I think Cantor would appreciate your sentiment that
>> the numbers of Cantor's paradise are more fundamental
>> than those of Kronecker's torment.

>
>
> I wouldn't say that infinity, even in the numbers, is either of those
> things. In ZF, Infinity is _axiomatized_ to be an inductive set, and
> a well-founded/regular one, that's not a given. Calling that the
> universe, Russell's comment is that it would contain itself.
>
> There's a case for induction, as it were, that each case exists. Then
> it is to be of deduction, not fiat by axiomatization, from simple
> principles of constancy and variety, the continuum.
>
> In a theory with sets as primary objects, a set theory and a pure set
> theory, numbers would be very rich objects indeed, as not just
> individual elements by their elements, but all relations of numbers.
> Set theory (well-founded, as it were, regular or that objects are
> transitively closed) is at once over-simplification, to talk about
> anything besides sets, and over-complexification, to talk about itself
> when any universal statement is in the meta.
>
> There are no numbers in a pure set theory. To call the natural
> integers a set, it contains only numbers, for the Platonists: elements
> of the structure, of numbers, as: none exist in a void.
>


It is odd. In some sense, modern mathematics actually
treats its objects as urelements relative to set theory.
Looking at Hilbert, he makes statements whereby his formalism
is intended to supersede the class-based constructions of
Dedekind.

Your frank statement that a set is not a number reflects
that sentiment.









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