Re: infinity

imaginatorium@xxxxxxxxxxxxx said:
> Tony Orlow wrote:
> > imaginatorium@xxxxxxxxxxxxx said:
> > > Tony Orlow wrote:
> > > > Virgil said:
> > > > > 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.
> > >
> > > Yes. Real set theory only talks about "completed" sets: a set is the
> > > collection of objects that are members of it. In the case of the set of
> > > naturals, it is the completed, total, entire, unexpandable set of all
> > > naturals. Yes, I know, you can't understand that.
>
> > And yet, any assumption of a largest element[1], an exact set size[2], a value range[3]
> > or mapping to the complete set[4], all cause contradictions.
>
> [1] Yes, because there is no largest element.
> [2] Yes, if by "exact" size you mean counting until the ditty ends, or
> measuring from one end (the one that exists) to the other end (the one
> that doesn't)
> [3] Yes, probably, but this "value range" stuff is your own rather
> ill-defined notion.
> [4] No, there is no contradiction in mappings involving all elements of
> a (total, completed, etc) infinite set. For example the mapping from x
> <-> 2x, for all pofnats x is totally well-defined.
Except for the fact that the top half of the naturals map to evens greater than
any element in the set of naturals. Ooops.

>
> > And you can't see
> > that that's because these sets don't end? We can measure them by applying a
> > variable endpoint, but ultimately, there is none.
>
> We have been trying to explain to you for months now, that infinite
> sets, such as the pofnats behave differently from finite sets precisely
> because they do not end. You have not exactly been the brighest student
> in the class, I might say. Note that it is only you who attempts to
> "measure" infinite sets by "applying" or "declaring" a "variable" (or
> "tenuous", or various other vaguenesses) end where there is none.
Only me? You mean, my idea is original? How delightful! Too bad original
thought doesn't count for much in the mechanistic world of today. Still, don't
expect me to stay in line.

They behave differently because they do not end. And yet, you want to include
an element that maps to it, after it has ended. Are you surprised that leads to
a contradiction? Neither *N nor its powerset end. The bijection continues. It
never fails. Bijections alone are not indicators of equal set size.
>
>
> > > > The y that maps to w at any point is outside of w, as can be plainly
> > > > seen with finite examples.
> > >
> > > No, you moron: y does not exist.
> > Yes, you idiot, in the infinite set, it does, dumbell.
>
> Unfortunately, this is elementary mathematics, and beyond you. Never
> mind.
> y is by definition an element of the set we are trying to map. It does
> not exist.
Is it coincidence that that is what you say about the largest finite? This
proof is exactly another of your menagerie of largest-finite proofs by
contradiction. If you can sneak in this assumption and derive a contradiction,
you can blame it on whatever the overt context of the proof is. It's a bogus
method of proof. There is a natural that maps to any idnetified set, and a set
that maps to any idnetified natural, and both sets are endless, so the
bijection is complete over the uneven infinite ranges.
>
>
> > > There cannot be an element that is
> > > mapped to the subset of exactly those elements not in the subset to
> > > which they map. There - the proof in one line. People have written it
> > > out carefully for you over many lines, and you just get lost. Put it in
> > > one line and I expect you will miss it.
> > This all assumes a given value range, which is not a mistake.
>
> Oh grief. You really are stupid, aren't you.

I am as stupid as someone who thinks the earth is round, or space is curved, or
the number line is round.

> Nobody but you has ever
> mentioned a "value range". WE ARE NOT TALKING ABOUT VALUE RANGES,
> whatever they are. We are talking about a set. The set of all of the
> topologically distinct polyhedra, for example. Even finite subsets of
> this set do not have a clearly defined "last member", so you can be
> sure the whole set doesn't. We do not consider a "value range" of these
> polyhedra, we consider the whole set of them.
And if they comprise an ordered set, then the power set is ordered too, and if
neither ends, then correspondence can be established on an order basis between
elements, if nothing else.
>
> Brian Chandler
> http://imaginatorium.org
>
>

--
Smiles,

Tony
.

Relevant Pages

  • Re: infinity
    ... >> Tony Orlow wrote: ... >>> You see the contradiction here? ... You have an infinite set, ... you proved that any consecutive sequence of naturals starting at 1 ...
    (sci.math)
  • Re: Would it matter if ZF was inconsistent?
    ... that avoids the contradiction found in ZF. ... his proof that PA is inconsistent -- meaning ... number N such that for all classical naturals n<=N, ...
    (sci.logic)
  • Re: infinity
    ... >>> Are you aware that if I claim I have a bijection from a set ... >>> elements from set A and maps them to elements of set B? ... half does not map to anything in the naturals. ... > do that by REASONING about all of the elements at once. ...
    (sci.math)
  • Re: infinity
    ... >> of 'infinite' is wrong? ... > set of naturals as defined by Peano is therefore infinite, ... you insist there is some contradiction. ... When talking about lists of positive integers, ...
    (sci.math)
  • Re: infinity
    ... >>> Not within the range of any given X, but in the overall infinite set, there is. ... >> elements from set A and maps them to elements of set B? ... it is not a mapping from A to B. ... > So, what happens to the top half of the naturals, in the mapping to the evens? ...
    (sci.math)
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