Re: Calculus XOR Probability

stephen@xxxxxxxxxx wrote:
Matt Gutting <tchrmatt@xxxxxxxxx> wrote:
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

But Tony's infinite sets do have maximal elements, such as
{1, 2, ... BigUn }. Tony is absolutely incapable of imagining a set without a maximal element. Everything must have two endpoints and be on
"the number line" or else he is simply incapable of understanding it.

Basically Tony like to use the word infinity, but he really does
does not understand or believe in the concept.

Stephen

But apparently even Big'Un isn't a maximum element, since you can go
beyond it by adding another unit infinity. It looks like the system has
"endpoints" which are functionally not endpoints (since you can surpass
them) but which are nevertheless declared to be so.

Matt

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