Re: infinity

In article <MPG.1dc6d45a6b9ec32a98a53e@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

> Virgil said:

> > TO asserts a priori that there are as 'many' digit positions in any
> > one of his strings as there are arbitrary binary strings in his set
> > of all strings. Since TO assumes that result a priori, his alleged
> > proof of it is circular and invalid.

> I assert that there is a never-ending supply of both natural numbers
> and subsets, so neither will run out.

TO seems to think that construction of a mapping is a process that
cannot be completed, but f:N -> N: x -> x^2 is a complete maping as
soon as it is written down.

In the same way, as soon as any mapping is completely specified, it is
complete.


> How is this different than assuming you have a never ending supply of
> naturals and evens, so you won't run out of evens before you run out
> of naturals?

TO seems to be assum that things like that x -> x^2 have to be
established one number at a time, and are never finished.

That may be true in the twilight zone of TOmatics, but is not true
anywhere else.




> > > > TO assumes, in his "construction" of that bijection that for
> > > > each infinite string of binary digits there is another such
> > > > string with containing one 1 and all other digits being 0. This
> > > > is clearly false.
> >
> > > Not clear to anyone outside of Virgilogic, so if Virgil wants to
> > > entertain his delusion, then I guess he'll be all alone!
> >
> > If it were true, why would anyone bother with the complicated form
> > of a number when a single non-zero digit is all that is needed? It
> > is only integral powers of 2 in any such binary notation that can
> > be represented by a single non-zero digit.

> Look, I don't even know what you think you said above, and this
> doesn't calrify anything, so it's not clear.

TO's limitations are showing!

> What are you talking
> about? I don't think I assumed whatever you think you said.
> >
> > To's assumption is requires, among other things, that every one of
> > the members of *N is an integral power of 2 (can be represented by
> > a single non-zero binary digit).

> Where did I assume any such thing?

You did not say it explicitely, but what you said imlpied it. TO is
quite blind t the implications of his claims, which is why his TOmatica
is in such a mess.


> You are confused. Take your red
> herring and go home.

It is TO who is quite confused about the consequences of his own
assumptions.

> >
> > TO also requires that his that for his f:*N -> P(*N), whatever it
> > may be, there exists some y in *N such that f(y) = {x in *N : x not
> > in f(x)}

> No, there can be no natural which maps to the whole set, since that
> depends on reaching the last element, which ain't there.

Then TO's mapping is NOT a bijection, as claimed.



> >
> > > It's better than the monkey cage! ;)
> >
> > The rest of us find TOmatics to be the monkey cage.
>
> Haha. You might want to note that the lock is on my side of the gate.

But the key is on our side and the monkey is safely locked inside
TOmatica and can't get out.
> >
> > > > The statement, and proof, of the theorem is solely about the
> > > > nonexistence of any surjection (therefore also of any
> > > > bijection) between any set X and its power set P(X). Neither
> > > > the theorem or its proof ever mention "size" in any other sense
> > > > than that of cardinality.
> > >
> > > The proof derives a contradiction from assuming some completed
> > > set and powerset.

There are no such things as uncompleted sets, at least outside TOmatics.
> >
> > The proof does not involve contradiction at all, unless one claims
> > what
> > cannot exist. It proves by construction that for any f:X -> P(X),
> > one can always construct a subset of X (and member of P(X)) not in
> > the image of f.

> You constructed no such set.

Given any f:S -> P(S), then T = {x in S:x not in f(x)} is a well defined
subset of S and member of P(S). The set constructs itself.


And anywhere outside TOmatics only "complete" are sets at all, so TO's
proces garbage is garbage.
.

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