Re: Cantor Confusion

mueckenh@xxxxxxxxxxxxxxxxx wrote:
MoeBlee schrieb:

mueckenh@xxxxxxxxxxxxxxxxx wrote:
Sorry, to say, but you always mix up these two very different things.
It should be comprehensible that potential infinity is possible without
the axiom of infinity. But that is not what set theory requires.
Therefore it is correct to say that in set theory theory there is no
infinity present or detectable without the axiom of infinity.

"Detectable", informally, okay. Yes, without the axiom of infinity, in
set theory (throughout all these discussions, by 'set theory' I mean,
in any given instance, some given Z or NBG variant) we cannot prove
there exists an infinite set. So, in that sense, we cannot "detect" the
existence of an infinite set. But we still are not permitted to claim
that infinite sets do not exist. We can only say that we cannot ever
detect whether they exist or not. You use these words such as 'is
present', 'detecable', etc., which are your words for your OWN way of
comprehending. But set theory is NOT in those terms that you use to try
to comprehend set theory.

Instead of your own METAPHORICAL language, let's look at the formulas
(or English renderings of actual formulas) of set theory:

Without the axiom of infinity, we cannot prove either of these two
formulas ('finite' and 'infinite' I define, respectively as
'equinumerous with a natural number', 'not equinumerous with a natural
number'):

Ex x is infinite

~Ex x is infinite

So without the axiom of infinity (but with no added axioms to the ZFC
axioms), we cannot prove:

Ax x is finite

To prove that formula, we need to adopt an axiom that entails it. Just
dropping the axiom of infinity does NOT entitle us to conclude Ax x is
finite.

Would you please just say whether you understand this point.

The following is correct:
There does not exist any actual infinity (i.e. what in modern set
theory is just called "an infinite set") neither in reality nor in
mathematics, unless you introduce the axiom of infinity. Only then an
actually infinite set exists in mathematics. Therefore, without this
axiom, everything and every set is finite.

Wrong. If we're talking about the axiom of infinity, then we're talking
about a set theory or some other theory you are welcome to specify. As
to set theory, for the tenth time: Without the axiom of infinity it is
UNDETERMINED whether every set is finite.

This includes sets without
bound which are called potentially infinite but which do not have an
infinite cardinal number.

In modern set theory you cannot say for all sets Ax x is finite. But
the negation of this statement is not what modern set theory calls
infinite.

In Z set theory:

~Ax x is finite -> Ex x is infinite.

I don't know what point you're trying to make.

Whatever your point, you won't be able to show that merely dropping the
axiom of infinity from the Z axioms entails that there are only finite
sets.

Modern set theory simply cannot describe developing sets as
it apparently cannot describe sets with limited contents of
information.

Whatever your definition of "developing sets" and 'limited contents of
information", the fact remains that dropping the axiom of infinity does
NOT entail that there are no infinite sets.

These things are unknown to the slaves of formalism. Read a good book
like Fraenkel, Abraham A., Bar-Hillel, Yehoshua, Levy, Azriel:
"Foundations of Set Theory", North Holland, Amsterdam (1984). There you
will find more about that topic.

Oh please, I've read more in that book than you have. But what is more
important, at least I know what that book is ABOUT, as opposed to your
unfamiliarity with the rudiments of the subject.

And Fraenkel, Bar-Hillel, nor Levy would never agree with you nonsense
that merely dropping the axiom of infinity entails that there are only
finite sets.

to look at the universe of all sets not as a fixed entity but as an
entity capable of "growing", i.e., we are able to "produce" bigger and
bigger sets. (p. 118)

Especially notice the scare quotes in that passage around "growing" and
"produce" (if those are scare quotes in the original quote). I highly
doubt that in context that is meant to be a claim that literally there
are sets in set theory that grows bigger and bigger (as opposed to our
option to adopt different definitions of a universe for set theory). I
highly doubt that the full contex of that quote permits it to be taken
literally that any given definition of a universe for a theory is not
of a fixed set or class. (I will look at the book again, and that
passage, next time I'm at the library.)

[Brouwer] maintains that a veritable continuum which is not denumerable
can be obtained as a medium of free development; that is to say,
besides the points which exist (are ready) on account of their
definition by laws, such as e, pi, etc. other points of the continuum
are not ready but develop as so-called choice sequences. (p. 255)

And there has been progress made even with proposals for intuitionistic
set theories. That doesn't alter that in Z set theories, dropping the
axiom of infinity does not entail that only finite sets exist.

MoeBlee

.

Relevant Pages

  • Re: Cantor Confusion
    ... contradiction there is in there not being such a mapping. ... That my proof is not indeed a proof in Z set theory. ... It is *assumed* by stating the axiom of infinity. ...
    (sci.math)
  • Re: Implementable Set Theory and Consistency of ZFC
    ... This is still not correct as the Axiom of Infinity is an axiom of ZFC ... Set Theory, even less when the latter is supposed to allow the Naturals ... consistent within common mathematics. ...
    (sci.math)
  • Re: Ultimate debunking of Cantors Theory
    ... Perhaps, for example, you think that specifying ... What axiom or theorem were you using to deduce ... exists without the axiom of infinity. ... thing about ZF set theory or how axioms are used to prove theorems. ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... most of the standard axioms would get scrapped ... one kind of set and infinity, and discrete counting systems form another, as in ... you claim that set theory is ...
    (sci.math)
  • Re: Cantor Confusion
    ... > definition leads the direct way to the result of actuallity we ... primitive language of set theory. ... As I have written again and again, you do not accept the axiom of infinity. ...
    (sci.math)