Agent Smith recently posted — alan1000

Do you mean his soul or spirit?

Agent Smith was banned like a year ago…

As a mathematician I don't recommend this topic for this forum. There are not many of us here with knowledge of transfinite theory. But maybe some interest will be sparked and the thread will be better populated than I imagine. :cool:

We're committed finitists here. Heresies are not allowed.

In all seriousness, IDK. I see a thread in the queue by a new user named an-salad? I assume it might be there due to lack of content; I think this post might already be longer than it.

We're committed finitists here. Heresies are not allowed. — Count Timothy von Icarus

Well, I think that there's something to Cantor's absolute infinity, and I've had splendid discussion about the topic here. And I've not been ban yet.

I think that there's just many issues in the fundamentals of mathematics that we don't understand yet. One thing is infinity, that set theory takes just as an axiom.

One thing is infinity, that set theory takes just as an axiom — ssu

Not all mathematicians are set theorists. "Without bound" works pretty well for some of us, without transfinite or philosophical overtones.

Yes, there limits etc. And then infinity as an axiom.

Yet if there's a Continuum Hypothesis, we clearly don't understand everything about infinity. Besides all the discussions about it that show it's not as obviously clear as some want it (or math) to be.

Yet if there's a Continuum Hypothesis, we clearly don't understand everything about infinity. — ssu

I think much of what we don't understand is a result of definitions in set theory. And when transfinite theory is incorporated into physics, practitioners take notice. I wonder if that has happened?

Anyhow, good luck in keeping this thing going. :smile:

. And when transfinite theory is incorporated into physics, practitioners take notice. I wonder if that has happened? — jgill

That's one important factor, actually.

Usually if we have something in math, it can be used well to model reality for example in physics. For me it tells that at least the math is correct.

That we don't have any use for the larger infinities in physics, at least yet, makes it doubtful that the Cantorian idea of larger and larger infinities is valid. After all, we've stuck with the question about the jump from the natural numbers to the reals.

What does it actually say?

A very interesting open question.

That we don't have any use for the larger infinities in physics, at least yet, makes it doubtful that the Cantorian idea of larger and larger infinities is valid. — ssu

I recall from my functional analysis courses the Hahn-Banach theorem, which deals with extending linear functionals on manifolds. This was the only time I encountered transfinitism. Even then a simple change in the hypotheses eliminated the need for going into multiple infinities. It's possible this theorem is the basis for a part of a mathematical process used in quantum theory. We'd have to ask an expert.

Add: Practical Transfinite

If something is very useful or practical in Physics, then I assume the math to be sound.

Faulty assumption. It could be that the physics is bad. Usefulness does not necessarily imply truthfulness.

If it's highly theoretical physics, then it might be so. But what works, it usually has some hang of the model. Yes, that might not be correct in general: hence if we would have stuck with Newtonian physics, I guess our GPS systems wouldn't be as accurate.

And I wouldn't say that a model/theory in physics is correct of false. It's usually either better or poorer.

I guess it depends on what you take the goal of science to be, usefulness or truthfulness. Traditionally, in "the scientific method", the ability to predict was taken as an indication of the correctness of an hypothesis. Now, it appears like many people believe that the capacity to predict is the goal.

Now, it appears like many people believe that the capacity to predict is the goal. — Metaphysician Undercover

That is the problem.

Hence you can have people working in science who say they don't care at all about philosophy.

Hence you can have people working in science who say they don't care at all about philosophy. — ssu

:smile:

It may not be easy to keep this thread from drifting away from the transfinite.

I have been looking into what Sir Roger Penrose has to say about transfinite theory in physics, and it appears he thinks it can arise in computational physics as that discipline evolves. He speaks of data fields and defines expressions like ${{\infty }^{r{{\infty }^{q}}}}$.

As for physics outside data fields he doesn't forsee transfinitism. As much as I have read.

Thank you everybody... when I posted the original question, which was really addressed to the moderators, I had no expectation of such a lengthy, interesting, and varied list of replies. I am surprised to learn that some mathematicians are inclined to be suspicious of transfinite theory, and question its value, because all of the philosophical problems of TF can be ultimately traced back, in logical sequence, to the 19th C Peano/Dedekind axioms of arithmetic - and if you are going to question those, then you can no longer assume that 1+1=2, and the entire science of mathematics falls on its butt in a crumbling heap!

As to different levels of infinity - the proofs that there must be at least two levels of infinity are so childishly simple, they can be understood even by high-school students who are not particularly bright in mathematics. The validity of the countable or aleph-null infinity is embedded in the properties of the natural number line. The validity of the aleph-1 or non-countable infinity is adequately demonstrated by Cantor's Diagonal Argument, which is very simple, and readily intellgible to non-mathematicians. Anybody who wishes to deny that higher-level infinities are valid in mathematical philosophy, must begin by pointing out where Cantor went wrong.

Are aleph-1 infinities useful in the sciences? Difficult question. We know that aleph-null infinities are not useful for investigating the origins of the universe, because every variable that we would wish to measure simply approaches infinity (or 0) as we approach closer to the singularity, and this of course is non-informative. Will the solution entail finding a way to apply the concept of the aleph-1 infinity? Some day, we'll know.

Does the 1st amendment extend to this philosophy forum? — alan1000

This is the First Amendment:

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Given that this philosophy forum isn't the United States Congress, the answer is no.

Also, as per the Terms of Service:

Except to the extent applicable law, if any, provides otherwise, this Agreement, any access to or use of the Website will be governed by the laws of the state of England and Wales.

Are aleph-1 infinities useful in the sciences? Difficult question — alan1000

Certainly non-rational mathematical objects, like e and pi, appear frequently in computations in physics, say. But they are always terminated at some point of calculation. Your question has relevance in the limit concept of calculus, of course. As computers are capable of approaching exactitudes they may compute further and further out on non-rational entities. Hence, it is best to have infinite expansions to consider.

:wink: Nice.

Physicist Sabine Hossenfelder has an interesting article about this:

Is Infinity Real?

The validity of the aleph-1 or non-countable infinity is adequately demonstrated by Cantor's Diagonal Argument — alan1000

Just to be clear: Cantor's showed that the set of real numbers is uncountable. He didn't prove that its cardinality is aleph_1. The assertion that the cardinality of the set of real numbers is aleph_1 is the continuum hypothesis, which Cantor did not prove, and which was later proven to be independent of ZFC.