As a connoisseur of cranks and sophists, I beg to differ. This thread is run of the mill in that regard. And there are routinely far more risible ignorance and confusion posted.

As for transfinite math, it rarely if ever comes up in classical analysis. — jgill

Depends on what is meant by 'transfinite math'. 'transfinite' is just another word for 'infinite', and, of course, analysis uses infinite sets. Moreover, there are mathematicians who work (and not in obscurity) with higher cardinals vis-a-vis analysis, though that work might not be prominent in the bread and butter mathematics you have in mind.

As for transfinite math, it rarely if ever comes up in classical analysis. — jgill

Depends on what is meant by 'transfinite math'. 'transfinite' is just another word for 'infinite', and, of course, analysis uses infinite sets. Moreover, there are mathematicians who work (and not in obscurity) with higher cardinals vis-a-vis analysis, though that work might not be prominent in the bread and butter mathematics you have in mind. — TonesInDeepFreeze

Here's how I see it for myself, transfinite math = Cardinals above the cardinality of the reals, or, treating infinities as objects. The only place this ever came up for me was a well-known theorem in functional analysis. Even there a slight adjustment in hypotheses removed its necessity.

Of course there are mathematicians who work with higher cardinals in analysis. They are at a higher level then bread & butter math. (there are still lots of questions in the latter, but the former is more attractive nowadays)

I'm not sure, but I think bread and butter analysis might touch on the cardinality of the power set of the set of reals (?), but I don't have enough information to dispute that even higher cardinals don't come up much.

But infinite sets are regarded as objects. The set of real numbers is a set theoretic object. Boom, from page 1 we are dealing with an infinite object. The real plane is the Cartesian product of the set of reals with the set of reals. Takes an two objects (or at least one object twice) to make a Cartesian product.

What is the domain of the function f where, for all natural numbers n, we have f(n) = 2*n? f is a function, and every function has a domain, and the domain of f is the set of natural numbers, which is an infinite set.

Even when we say "let n go from 0 to inf", that really is just to say that the domain of the function is the set of natural numbers.

I don't see the point in saying that mathematics such as analysis doesn't use infinite sets, when plainly, at the very outset, to even start in the subject, we see that we are using infinite sets.

How would a difference in size be established between two infinite sets when there is no counting involved?

Same way we determine a set is infinite without counting it: stipulations.

We cannot determine that a set, S, is infinite by counting the elements (as we would never be able to stop, and this doesn’t discern a set that is indefinite from one that is infinite). Instead, we could determine S is infinite either by stipulation—e.g., if we are considering the set of all natural numbers, then we thereby know that this set is infinite because there is an infinite amount of them.

Likewise, we cannot determine that S1 is larger than S2 by counting the elements; instead, we come to know it by understanding the stipulations of the sets themselves. If S1 is a set with size 2 elements ad infinitum and S2 is a set with size 1 of elements ad infinitum, then S1 > S2 (and I don’t need to count them).

I'm not sure, but I think bread and butter analysis might touch on the cardinality of the power set of the set of reals (?), but I don't have enough information to dispute that even higher cardinals don't come up much. — TonesInDeepFreeze

I have explored a topic in classical complex analysis over the years. It is not a popular topic and many of those initially interested have passed away. I have written close to a hundred articles and notes, about a third of which I published before retiring in 2000. After that, publishing was too much a hassle; shorter notes on researchgate.net . None of them use the power set of the set of reals.

You may not know how many topics there are in math. Wikipedia has, I recall, about 26K pages. When I open a page at random I usually am clueless about what I find. ArXiv.org gets over a hundred math research papers a day, listed in various general categories. Even in classical complex analysis, I usually am left behind.

The output of mathematicians is staggering. However, it seems to me there used to be either a category for Set Theory or Foundations in ArXiv.org . It's no longer there. There is one for Logic, and this title caught my eye: Mice with Woodin cardinals from a Reinhardt

I have mentioned before that I am old and outdated. Not a reliable authority for TPF.

I don't see the point in saying that mathematics such as analysis doesn't use infinite sets, when plainly, at the very outset, to even start in the subject, we see that we are using infinite sets. — TonesInDeepFreeze

Analysis normally does not dwell on set theory. It's there in the background of foundations. And the limit concept arises from it, but when I use limit, as defined using epsilon/delta, I don't go into set theory details. If I say x->1 it is assumed it does so through the reals.

we could determine S is infinite either by stipulation—e.g., if we are considering the set of all natural numbers, then we thereby know that this set is infinite because there is an infinite amount of them. — Bob Ross

That is circular.

And we don't just stipulate that the set of natural numbers is infinite. We prove it.

If S1 is a set with size 2 elements ad infinitum and S2 is a set with size 1 of elements ad infinitum, then S1 > S2 (and I don’t need to count them). — Bob Ross

It's nothing like that.

Right, those working in the various branches don't usually concern themselves with the foundations. So use of infinite sets is ubiquitous without concern for the foundational axiomatization concerning them. Proverbially, infinite sets are the water mathematics swims in. The fish doesn't have to know anything about water, but it still needs that water to swim in.

As I argue, the first day of class when we are told "We have the natural numbers and we have the real numbers and the real number line", boom, we are presented with infinite sets, even if the instructor doesn't happen to mention, "And don't forget, those are infinite sets".

As to age, the mathematics is pretty durable, thus also is the wisdom of those who learn it.

How's it circular? Demonstrate where I am begging the question.

'S is infinite' is equivalent with 'S has infinitely many members'.

Or as you say:

if we are considering the set of all natural numbers, then we thereby know that this set is infinite because there is an infinite amount of them. — Bob Ross

'the set of natural numbers is infinite' is equivalent with 'there is an infinite amount of natural numbers'.

Proving that the set of natural numbers is infinite is the same as proving that there are infinitely many natural numbers.

In a textbook in set theory, you would see how a theorem of the form:

S is infinite

is actually proven.

What is the proper interpretation of the cosmological constant Λ? I understand that it corresponds to a vacuum energy density, pervading all reality. Such energy is called dark energy, I gather. Since I'm sketchy on field theory, I don't know how this goes, but somehow this energy density produces a repulsive force beween any two objects in spacetime (within each other's lightcones?). Matter remains cohesive because Λ is very small compared to other forces, so that its effects really only show at an intergalactical scale (megaparsec). — DanCoimbra

I would say that is one interpretation of "dark energy", but here would be a number of possible interpretations.

ow, somehow this leads to the expansion of the Universe even in the case where the Universe is finite and bounded, which is a possibility considered by cosmologists. In this case, the Universe is increasing in total size, but not increasing *into* anywhere, so it becomes bigger because it has more internal spatial structure. This is what I meant. Why do you think this is incorrect? — DanCoimbra

What I meant is that "more internal spatial structure" is not consistent with Einsteinian relativity, because that would render a whole lot of predictions about the motions of things as inaccurate. We can posit "dark energy" as the reason why the predictions are inaccurate, but then where is this dark energy, and what is it doing other than making the predictions inaccurate,
.

I just think if mathematical axioms are to be selected, they have to be such that they do not lead to what is contradictory to Existence/Truth (or just semantics in general). — Philosopher19

The problem inherent within pragmaticism is that whatever is the purpose at the time (the flavour of the day), the axioms chosen will support that purpose. As time goes by, and needs change, other axioms will be produced to satisfy the evolving needs. At this point, the new and the old are not necessarily consistent, so there may be a degree of contradiction between different logical structures, depending on the purpose which they each serve,

If a mathematician or a philosopher decides on an axiom or theory that requires belief in the following (or at least logically implies it or leads to it): Nothing can be the set of all things (which logically implies Existence is not the set of all existents), or one infinity is a different bigger than another (or is a different quantity than another), I believe that axiom or theory should be disregarded or at least viewed as contradictory to Existence/Truth (or at least contradictory to the semantic of infinity). — Philosopher19

I agree that it is appropriate to set a standard of "truth" for mathematical axioms.

if each cardinal is STRICTLY larger than the one before it, I suppose they do indeed have different sizes. — Vaskane

By definition, a successor cardinal is strictly greater than its predecessor.

(By the way, aside from successor cardinals, there are cardinals other than 0 and aleph_0 that are not successor cardinals, and they are greater than any previous cardinal.)

Anyway, without even getting into successor cardinals and limit cardinals (cardinals that are not 0 and not successor cardinals), it is easy to prove that for any set, there is a set of greater cardinality.

[set theory says] Nothing can be the set of all things (which logically implies Existence is not the set of all existents) — Philosopher19

Mathematics doesn't mention "all existents" or "set of all things".

The heart of your attack on infinitistic mathematics is your own mistaken fabrication of what you think the mathematics is. In other words, you're putting up a huge strawman.

If so, then you understand that a line of an interval of 2 represent twice the length, as the line of an interval of 1. And thus you're perhaps an even lower wisdom score than 8 after I already pointed out several times that there's an error in communication and even held myself accountable for that error, that you're too dumb to understand a line has length/area/size whatever the fuck you wanna call it, after I clearly stated a communication error upon the context ... I mean fuck dude, you're like Marine when he sees red. — Vaskane

As that was added in edit, I missed it.

Whatever you think of me, or whatever error you think there was in communication, I accurately responded to your posts as they were written.

You claimed that the size of the set of numbers between 1 and 2 is less than the size of the set of numbers between 1 and 3. If that's not what you meant, then it's not my fault. Then you deflected to the fact that the distance between 1 and 2 is less than the distance between 1 and 3, which is true, but it does not bear on the fact that the size of the sets is the same. At the time of posting I saw no post in which you "held yourself accountable" for that error.

And I explained in perfect detail about length, but instead of recognizing that, you incorrectly suggest that I don't understand length and you resort to juvenility such as "too dumb".

It makes a real difference. By saying 'infinity' as a noun and then that there are different sizes of infinity is to picture an object that has different sizes. There is no such object in mathematics.
— TonesInDeepFreeze

I don't think I'm picturing an object. I think I'm just focused on the semantic of Infinity. — Philosopher19

Whether described as 'picturing an object' or 'positing that there is such an object' my point is that set theory does not mention, describe or posit any such object, so saying 'Infinity' as a noun as you do is misleading as it does suggest that one should take set theory as suggesting that such an object can be countenanced, considered or pictured, etc.

Good faith in posting a critique of mathematics would entail at least knowing something about it.
— TonesInDeepFreeze

I think it is from all that I have seen and heard [...] — Philosopher19

What are your sources? What specific texts in set theory or mathematics do you think have said the things you claim set theory to say?

Whether all that I have seen or heard is enough, is another matter. You don't think I have. I think I have. — Philosopher19

It's the heart of the matter of why your are ignorantly misrepresenting set theory.

You think you've read enough set theory to understand its axiomatic treatment of infinite sets? What specifically have you read, let alone studied sufficiently to competently discuss it?

There is no object called 'Infinity' in the sense you have been using it.

Here is a way to say what you want to say:

In mathematics, there are sets that are infinite but that have different cardinality from one another.

Better yet:

If x is infinite then there is a y that is infinite and y has greater cardinality than x. — TonesInDeepFreeze

But all of the above is exactly what I'm saying is contradictory. And my use of infinity which (if I've understood you correctly) you say is not the one that they use in maths, is the reason that I say all of the above is contradictory.

There is no object called 'Infinity' in the sense you have been using it. — TonesInDeepFreeze

So what semantic are mathematicians using when they use the world/label "infinite"?

There is no x such that for all y, y is a member of x iff y is not a member of y. Proof: — TonesInDeepFreeze

the axiom schema of separation — TonesInDeepFreeze

Something cannot be both a member of itself and a member of other than itself at the same time. For example, take z to be any set that is not the set of all sets, and take v to be any set. The z of all zs is a member of itself as a z (as in in the z of all zs it is a member of itself). But it is not a member of itself in the v of all vs, precisely because in the v of all vs it is a member of the v of all vs as opposed to a member of itself. If we view the z of all zs as a z, it is a member of itself. If we view the z of all zs as a v, it is a member of the set of all sets. You can't view it as both a member of the z of all zs and a member of the v of all vs at the same time. That will lead to contradictions. In other words, we can't treat two different references as one (as in are we focused on the context of vs or the context of zs?)

Note that the above shows the impossibility of a set that contains all sets that are members of themselves where all equals more than one.

For the fully fleshed out version of this, see my post on Russell's paradox which I posted the link to in this discussion and in the other one.

'There exists a z such that for all y, y is a member of z' contradicts this instance of the axiom schema of separation: For all z, there is a x such that for all y, y is a member of x iff (y is a member of z and yis not a member of y). — TonesInDeepFreeze

If some theory suggests that you can view the z of all zs as both a member of the z of all zs and a member of the v of all vs at the same time, then that theory is contradictory. The z of all zs is either to be treated like a z or a v. If it is to be treated like a z, it is a member of itself. If it is to be treated like a v, it is not a member of itself (precisely because it is a member of the v of all vs)

We do NOT claim that from "after each natural number there is a next number" and "there is no greatest natural number" that we can infer that there is a set of all the natural numbers. Indeed such an inference IS a non sequitur. And every mathematician and logician knows it is a non sequitur. So, we recognize that to have a set with all the natural numbers we need an AXIOM for that, which is NOT an inference. — TonesInDeepFreeze

I'm not sure what you mean by "So, we recognize that to have a set with all the natural numbers we need an AXIOM for that, which is NOT an inference."

We go in a circles, as it is with cranks. The crank makes false claims and terrible misunderstandings. Then the crank is corrected and their error is explained. Then the crank ignores all the corrections and just posts the false claims and misunderstanding again as if the corrections and explanations never existed.

So it is again regarding 'contradiction'.

Go back and read what I wrote about "contradiction".

Something cannot be both a member of itself and a member of other than itself at the same time. — Philosopher19

In certain alternative set theories, there are sets that both members of themselves and of other sets.

In ordinary set theory, no set is a member of itself.

By the way, we don't need to use temporal phrases such as "at the same time". Set theory does not mention temporality.

Then the rest of your z's and v's is irrelevant if it is supposed to refute the proofs I gave. Moreover, if you knew anything about this subject or even mathematical discourse you'd see that your prose about it is ungrounded, impenetrable double-talk.

To refute a purported proof, you need to show a step in the proof that is not permitted by the inference rules (which in this case are those of ordinary predicate logic).

And you separately quoted me saying "The axiom schema of separation". What was the point of that? Did you look up what the axiom schema of separation is and you think your remarks relate to it in some way?

I'm not sure what you mean by "So, we recognize that to have a set with all the natural numbers we need an AXIOM for that, which is NOT an inference." — Philosopher19

It is clear. You don't know what it means, because you are virtually completely ignorant of the subject matter.

I'll spell it out even more:

In set theory, to prove there exists a set having a certain property, we must do so from the axioms and rules of inference alone.

In this instance, the property in question is "has as members all the natural numbers"

Without the axiom of infinity, we cannot prove that there is a set with the property "has as members all the natural numbers". But with the axiom of infinity we can prove that there is a set with the property "has as members all the natural numbers".

You argued that from "after each natural number there is a next natural number and there is no greatest natural number" we cannot infer "there is a set of all the natural numbers". And you are correct about that!

So I pointed out that indeed set theory does not make that unjustified inference, but rather, set theory has an axiom from which we CAN infer that there is a set of all the natural numbers. And THAT inference, from the axiom, does NOT use the unjustified inference from "after each natural number there is a next natural number and there is no greatest natural number" to "there is a set of all the natural numbers".

/

You know virtually nothing about set theory. You should present whatever concept of infinity you like, but you shouldn't be presenting it as a refutation of a subject you are ignorant about.

You know, its a funny thing, but when I don't know much about a subject, I pay attention to people who do know something about it. And especially I don't slather the Internet with stubbornly false and confused claims about it.

So what semantic are mathematicians using when they use the world/label "infinite"? — Philosopher19

We don't say "using semantic".

Rather, we just state the definitions.

I stated the definitions in my first post in this thread:

https://thephilosophyforum.com/discussion/comment/878326

So what semantic are mathematicians using when they use the world/label "infinite"? — Philosopher19

Mathematicians, like myself, may get a little sloppy about using the word, infinity, at times. For example, for those of us in complex variable theory The point at infinity has a specific reality as the north pole of the Riemann sphere. There is a technical way of saying this.

I have an ability to understand concepts without even knowing of them — Vaskane

Please forgive the cliche, but it is especially apt: Above is Dunning-Kruger on steroids.

Here is a finite definition of an infinite set: "A given set S is infinite iff there exists a bijective function between S and a proper subset of S." Furthermore, such a bijective function can be stated finitely.

Here is an example. Take the set of natural numbers ℕ = { 0, 1, ··· }. Now take a proper subset of ℕ containing only even the numbers, ℙ = { 0 , 2 , ··· }. These two are equinumerous because there is a bijective function f : ℕ → ℙ, given by f(n) = 2n.

The proof that "f" is bijective is finite. So is the proof that ℙ is a proper subset of ℕ. — DanCoimbra

Great post, thanks. How do you prove then N is different size to P?

Did you give Philosopher19 the finger or is there real math behind the north pole of the riemann sphere? It would be cool if you meant it both ways.

In certain alternative set theories, there are sets that both members of themselves and of other sets. — TonesInDeepFreeze

We go in a circles, as it is with cranks. The crank makes false claims and terrible misunderstandings. Then the crank is corrected and their error is explained. Then the crank ignores all the corrections and just posts the false claims and misunderstanding again as if the corrections and explanations never existed. — TonesInDeepFreeze

Evidently, there's no point in continuing this discussion. If you believe your mathematics is free from contradictions or paradoxes, then in my opinion, you are not blameworthy for upholding them or sticking to them (unless of course someone presents a better or more complete thing to you and you reject greater for lesser), but if you see paradoxes and contradictions or incompleteness and you treat them as other than paradoxes/contradictions/incompletions...
I see no paradoxes or contradictions or foundational incompleteness in the beliefs that I uphold (mathematical or otherwise).

Peace

I will just say this. That a set cannot be both a member of itself and a member of other than itself is the equivalent of saying that a shape cannot be both a square and a triangle (I have taken out the "at the same time" and the effect is still the same).

The above point I felt was worth adding to this discussion, but I will probably stop posting here as I don't think there's anything left to add to this discussion.

This isn't really the place to come to get people to agree with you. I think the math boys really did give you a good amount of feedback that would be hard to get anywhere else. So if you want to run something past us we'll tell you what we think and you can react accordingly. Most of what you say really irks a formally trained mathematician.

To me it seems like arguing about mental fantasies but for someone who has studied it there would be something to defend.

It's been one of the more lively threads here...seems to go on all day.

As far as the math profession I do think you should show some respect because the world runs on the math they do and for some things only a few people per million or billion may be able to do it.

This isn't really the place to come to get people to agree with you. I think the math boys really did give you a good amount of feedback that would be hard to get anywhere else. So if you want to run something past us we'll tell you what we think and you can react accordingly. Most of what you say really irks a formally trained mathematician. — Mark Nyquist

Here's actually some advice to all non-mathematicians (from a non-mathematician):

If you really can ask an interesting foundational question that isn't illogical or doesn't lacks basic understanding, you actually won't get an answer... because it really is an interesting foundational question!

Yet if the answer is, please start from reading "Elementary Set Theory" or something similar then yes, you do have faulty reasoning.

How do you prove then N is different size to P? — Corvus

We don't. He proved that they are the same size.

We go in a circles, as it is with cranks. The crank makes false claims and terrible misunderstandings. Then the crank is corrected and their error is explained. Then the crank ignores all the corrections and just posts the false claims and misunderstanding again as if the corrections and explanations never existed.
— TonesInDeepFreeze

Evidently, there's no point in continuing this discussion. — Philosopher19

As I said, the discussion will go in circles given that you skip answers given you and instead just repeat your refuted claims.

If you believe your mathematics is free from contradictions or paradoxes — Philosopher19

It's not my mathematics. I don't have allegiance to it. I find value in it, find wisdom in it, recognize that it axiomatizes reasoning used in the sciences, and enjoy it. But I don't claim that there might not be better approaches - philosophically, intuitively, and practically.

I don't claim to perfect certainty that set theory is consistent. But it seems to me to be an extremely good bet that it is. (1) No contradiction has been found in it under incredibly intense and indefatigable scrutiny for about 125 years. (2) We can see specifically how it was devised to avoid Russell's paradox. (3) The concept of sets as a hierarchy itself suggests an intuitive approach that is consistent.

Again, you use the word 'incompleteness', thus ignoring the information that was given you about incompleteness.

I see no paradoxes or contradictions or foundational incompleteness in the beliefs that I uphold (mathematical or otherwise). — Philosopher19

You haven't proposed an alternative framework, let alone in axiomatic form. Articulate the principles by which you propose to derive mathematics adequate for the sciences, or, better yet, put it in axioms; then we can put it to the test.

Set theory gets the job done of axiomatizing the mathematics for the sciences. By analogy: Set theory is an airplane that flies. If one thinks it's not a good airplane, then one is welcome to show us a better one.