Re: an important set theory post

On Jul 23, 3:15 pm, amy666 <tommy1...@xxxxxxxxxxx> wrote:
i wonder why lwalke didnt reply ...

I've been very busy lately and haven't had much
time to post lately. Meanwhile, this thread has
exploded especially since there are in fact two
people challenging ZFC in this thread. So let's
see whether I can catch up here.

Rereading tommy1729's long post, we see that
the root of the problem is that he accepts the
existence of cardinals but not the existence of
ordinals at all. But the alephs are indexed by
_ordinals_, so one can't talk about alephs --
especially the alephs that are greater than the
first infinite singular cardinal -- without
mentioning ordinals.

Let me quote from tommy1729's long post and I
will comment on them using ZF+GCH:

Actually, one can take the powerset as many times
as we wish, countable or countable, in ZF+GCH.
yes but is oo countable = oo uncountable in this context ?

I meant uncountable here. This is my mistake, a
typo, so let me correct it here.

thus is aleph_aleph_aleph_0 = aleph_aleph_1 or not ?

They aren't the same. Since aleph_aleph_0 is not
aleph_1, aleph_aleph_aleph_0 is not aleph_aleph_1.

and if uncountable repetition of the powerset gives it ,
what " is " this uncountable repetition of the power set ?

It means that we take the Powerset once for every
countable _ordinal_ up to aleph_1. Since tommy1729
doesn't believe in the existence of _ordinals_,
small wonder he doesn't accept this construction.

cantor said P(x) > x
clearly aleph_aleph_0 already violates that.
in fact it is russels " set of all sets " !!

Actually, ZF+GCH proves that P(aleph_aleph_0)
has cardinality aleph_(omega+1). Since tommy1729
doesn't believe in the existence of _ordinals_,
small wonder he doesn't accept this construction.

thus aleph_aleph_0 = aleph_aleph_0 + 1
but aleph_omega =/= aleph_omega + 1.

And here tommy1729 has unwitting demonstrated
exactly why we use _ordinals_ instead of
_cardinals_ to index the alephs -- since we
_want_ the successor cardinal of aleph_aleph_0 to
be distinct from aleph_aleph_0. Since tommy1729
doesn't believe in the existence of _ordinals_,
small wonder he doesn't accept this construction.

yet 2^aleph_aleph_0 = aleph_aleph_0 + 1 = aleph_aleph_0
( and not aleph_aleph_1 as many here incorrectly claim !!!!!!!!!!!!!! david & denis are subsets of many as anticipated )

Neither David Ullrich, Denis Feldman, nor myself
have claimed that 2^aleph_aleph_0 = aleph_aleph_1
in ZF+GCH. Instead, the claim is that
2^aleph_aleph_0 = 2^aleph_(omega+1). (Of course,
it may be possible in a model of ZF+~GCH that
2^aleph_aleph_0 = aleph_aleph_1 -- I'm a bit
hazy on the cofinalities of the sets involved.)

nobody in the world can explain this (in)equation :
2^aleph_x = aleph_x+1 =/= aleph_1+x

I can explain it -- but the explanation requires
the use of _ordinal_ numbers. Since tommy1729
doesn't believe in the existence of _ordinals_,
small wonder he doesn't accept this construction.

lets use 2^aleph_x = aleph_x+1 and brackets to clarify ;
2^aleph_(aleph_0) = aleph_(aleph_0 + 1) = aleph_(aleph_0)

Not if one uses _ordinals_ to index the alephs.

similarly 2^2^2^aleph_aleph_0 = aleph_aleph_0

Not if one uses _ordinals_ to index the alephs.

so the " tetration " seems to converge from the very first application already ( 2^ ) and even if applied any finite amount of times ( 2^2^2^ .. ) or even

Perhaps I was doing more harm than good by mentioning
that old thread about cardinal tetration. In that
thread I didn't want to use _ordinals_ since we were
trying to define _cardinal_ tetration. But the alephs
indexed by ordinals provable exist in ZFC, with or
without a definition of tetration.

it would then be IMMORAL to add axioms for aleph_aleph_1 , why then not allow aleph_-1 too hmm ?
dropping conditions to your own pet theory benefit hmm ?

Because aleph_aleph_1 provably exists in ZFC, with or
without a definition of tetration, and without adding
any more axioms. But aleph_-1 would require some sort
of nonstandard set theory.

Notice how both calvin and tommy1729 reject the use
of infinite _ordinals_ to index the alephs. Thus
the comments made by standard mathematicians in
response to one of them equally apply to the other.
.

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