Re: Calculus XOR Probability

Matt Gutting said:
Tony Orlow wrote:
MoeBlee said:
Tony Orlow wrote:
MoeBlee said:
Tony Orlow wrote:
To say
all elements are finite but the set is infinite implies that there is some
lartgest finite with an infinite successor.
In what theory? In set theory? No, what you said is not true of set
theory. So in some other theory? If in some other theory, then what are
its axioms and primitives and what are its definitions of 'infinite',
'largest', 'finite', and 'successor'?
I was speaking of the logic behind the limit ordinals, and how it is flawed.
You said "To say all elements are finite but the set is infinite
implies that there is some lartgest finite with an infinite successor."
That statement contradicts set theory. So my question, which you did
not answer, is: In what theory do you cliam that your statement holds?

And there is no flaw iin the logic behind limit ordinals. The only
logic involved in set theory is first order predicate logic.

I suppose what it boils down to here is infinite induction. Since the size of
the set is the successor to the maximal element in all finite sets, this
relationship should hold, being an equality, in the infinite case. If that is
so, then omega is successor to the largest finite, and the notion is self-
contradictory. Infinite induction appears to be discredited, but the only
counterexample in this thread is easily explained otherwise, and "infinity did
it" doesn't fly when it comes to explaining an error of sqrt(2). The limit
ordinal omega is in direct contradiction with infinite induction, so one of
them is wrong. Hint: it's not infinite induction.

The problem I see with this is that the finite cases, as you yourself
point out, deal with sets having maximal elements. Since this is not true
for "the infinite case", I don't see how you can apply the same reasoning
there.

Matt

*** Posted via a free Usenet account from http://www.teranews.com ***


What does that have to do with anything? This is an inductive proof that for
every finite natural, the number of naturals up to that point is finite. Does
that not mean that there is no point at which the set of finite naturals is
infinite? If it does not become infinite within the range of elements that it
includes, how is it infinite? By including things outside the set?
--
Smiles,

Tony
.

Relevant Pages

  • Re: infinity
    ... Not while discussing the consequences of the axioms as they actually are! ... > You never commented on my adjustment of the Peano axioms, ... IF you claim to have generated an infinite set with your ... I am applying only finitely many successor operations at ...
    (sci.math)
  • Re: infinity
    ... >>> The Peano axioms prohibit any complete Peano set from having a largest. ... You never commented on my adjustment of the Peano axioms, ... Well, after all, you ARE applying an INFINITE number of successor operations, ... IF you claim to have generated an infinite set with your stepwise difninition, ...
    (sci.math)
  • Re: Galileos Paradox and the Project of the Reals
    ... Can you think of any model for that axiom system, ... "successor" function on that set, ... only an infinite sized set. ... successor facts ), even if every element in the set already had ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... all elements are finite but the set is infinite implies that there is some ... lartgest finite with an infinite successor. ... implies that there is some lartgest finite with an infinite successor." ...
    (sci.math)
  • Re: Infinity
    ... >>> there are some set theory conclusions regarding infinity which indicate some ... >> The definition of 'Dedekind infinite' is: ... now you've made it 'distinct successor'. ... the proof of the Banach-Tarski theorem does not use ...
    (sci.math)