Re: Cantor's diagonal proof wrong?
From: George Greene (greeneg_at_cs.unc.edu)
Date: 11/23/04
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Date: 22 Nov 2004 23:44:18 -0800
raf@tiki-lounge.com (Ross A. Finlayson) wrote in message news:<3c6b9c1e.0411180948.d9288be@posting.google.com>...
> Hi,
>
> Basically from the empty set all possible constructions, which means
> basically any group of balanced brackets, is a set.
That doesn't mean jack.
You have to provide some axioms for how to tell when/whether
an INFINITE string of brackets is balanced. Otherwise,
this only defines finite sets. That is not exactly interesting
as a universe.
> As well, as there is no foundation axiom,
Is there an anti-foundation axiom?
Are you going to explicitly assert the existence of some infinitely
deep sets? Just "allowing" it is hard because the standard
definition of a first-order language does NOT allow infinitely
deep terms.
> then you would be able to arbitrarily label a
> set and then insert that label anywhere within that or any other set
> definition.
Allowing labels at all is, again, an ADDITION to the paradigm
of a traditional first-order language. You can't just handwave
that. You have to talk about what wff's WITH labels (as opposed
to without) LOOK like. You have to define YOUR terms.
> Each of these is formed because each is unique, and thus through
> excluded middle, not your axiom of difference, it is generated.
You don't know jack about what "excluded middle" can "generate".
In point of actual fact, at the theoretical level, the middle
is NOT excluded. Everything that hasn't yet been proved or disproved
is STILL in the middle, and some things, things that can by meta-
theoretical and model-theoretical means be PROVEN independent of
the axioms, are LOCKED into the middle.
> The sets by themselves can be quite meaningless.
In the traditional paradigm, the content of the universe is
irrelevant; any other isomorphic batch of content would serve
as well.
> It's just the set of all possible constructions of the empty set,
> the set of all sets.
THAT, on the other hand, IS meaningless.
>
> Then, each set is claimed to fall together in a consistent way to form
> ordinals. Take any literal,
You don't know what a "literal" is.
Please note that the rest of the world's set theories have
no such notion, so if you are going to invent one, you FIRST
need to DEFINE it.
> So there aren't any axioms, just an assertion of empty set
How exactly do you propose to assert "the empty set exists"
WITHOUT an axiom????
> and excluded middle,
Excluded middle is part of the first-order paradigm.
It does not need to be asserted in the object-language
and it does not "generate" anything.
> and a definition of ordinal.
Again, how do you propose to have a definition of anything
WITHOUT some axioms???
- Next message: Sue Turner: "R/S System: Advanced Programming, San Francisco***December 20-21"
- Previous message: Eray Ozkural exa: "Re: the modesty of Mr Wolfram. On the Foundation of mathematics and mathematica"
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