You are not a very well informed person - finitism is quite a widely held view and quite a broad church:

https://en.wikipedia.org/wiki/Finitism

Lots of people doubt the reality of actual infinity. It just happens that not so many finitists frequent this site.

Actual infinity means in a strict mathematical sense a number greater than any other. But there is no greatest number. So there is no actual infinity.

Actual infinity in a physical sense means something that goes on forever. But only in our minds can things go on forever / be infinity dense etc... these things are impossible in reality.

So that implies something false can be added to set theory without changing its consistency. — Devans99

Yes. Consistency and truth are not the same thing. That is one of the main consequences of Godel's incompleteness theorems https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

Truth is about the interpretation of a proposition on a concrete model. I a not even sure if there is a commonly accepted definition of what is a model, or a widely accepted definition of truth. I believe the definition of truth is more in the domain of philosophy: working mathematicians treat their (abstract) mathematical models as if they were real concrete physical models and the propositions were concrete physical experiments on which a proposition can be tested to obtain a result true or false.

Consistency, on the contrary, is defined as a property of a formal logic system: a formal system is inconsistent if a proposition can be proved to be both "true" and "false". "true" and "false", in this case, is are purely syntactical objects: a string of characters that is the result of the execution of an algorithm.
If a proposition cannot be proved to be "true" nor "false", it (it's interpretation as experiment, let's say) can be true for some models and false for other models. If a proposition can be proved, it has to be true for all models. It it's negation can be proved, it has to be false for all models.

For example, if I define a simple maths system with only one number: 1 and one operator: + then I can axiomatically define 1+1=1. Its consistent but not logical. — Devans99

I believe that the word "logical" is really used as a synonym of "consistent", but only in colloquial english. In mathematics I think it's not used at all.
A simple math system may be consistent even if it doesn't have any model (except from the "purely syntactical one": every consistent logical system has a model made regarding as objects of the model the syntactical symbols of the language). You can prove the proposition "1+1=1", but you cannot say that it's true if you don't specify in which model. (By the way, a formal logic system is not a completely arbitrary set of rules an symbols: symbols of negation true and false have to be defined: otherwise the definition of "consistency" itself makes no sense)

P.S. Sorry, I just realized that what I wrote about the definition of consistency is wrong: a formal system is inconsistent if for some proposition both the proposition and it's negation can be proved. I think the strings "true" and "false" (or something equivalent) are not required to be part of a formal logic system (well, everything depends on which logic we are speaking about: there are too many of them.. :-) ).

Truth is usually defined in philosophy as justified true belief. I guess this definition of truth must refer to what is true in the real world by default. It could refer to truth within some abstract axiomatical system also, but I think the first usage is the common usage.

Any truth in an axiomatical system is only as true as it's axioms (and the axiom of infinity in set theory seems false) whereas true in the real world means more than that - it means it fundamentally reflects the way the real world is.

So set theory or any axiomatical system can be consistent but not true at the same time. But it is (or was) the habit of mathematics, physics, logic to choose axioms that are very likely true in the real world. IMO set theory has an axiom that is very likely false in the real world. It should come with some sort of health warning attached because it has confused a lot of people into believing that actual infinity is a physical reality.

There are other axiomatical system that may or may not be true in the real world. For example non-euclidean geometries. But set theory stands out as choosing an axiom that is clearly false in the real world - so it has no real world applications because of that (the part of set theory relating to infinite sets I mean, finite set theory is useful).

I feel that axiomising something that is clearly false in the real world (IMO) is the basis of my beef with set theory.

I feel that axiomising something that is clearly false in the real world (IMO) is the basis of my beef with set theory. — Devans99

I feel this is putting what you said slightly better. When we expand on mathematical concept with new information we can produce new axioms. The problem with infinite is that it is polymorphic word or concept. When you use infinite in one statement it might have slightly different meaning then if used in a different context. To expect the same axioms to apply to each different (polymorphism) situation is not mathematically or logically sound. I don't know if you stated what i just said previously. Sorry if i duplicated information.

It is true there are different definitions. Maths has a definition relating to a one-to-one mapping between a set and a proper subset that is quite odd and different to the normal usage. Personally I use Aristotle's definitions of actual and potential infinity. A common usage is something bigger than anything else possible.

Whatever the exact definition, actual infinity leads to problems... how can a subset be the same 'size' as its containing set? The whole is greater than the parts. How can something be bigger than anything else possible? How can something go on forever? IMO we are in the realm of make believe if we countenance these... unicorns, pixies, magic and actual infinity are all of the same mould.

The problem is certain concepts such as infinite have not been systematically defined for different contexts. In software development you will use the same function or method in different ways simply by changing the arguments that go into that function or method. (the method can be defined with in the program completely differently by the programmer and the compiler for example will know which one to use simply by the arguments or context changing). Infinite and other concepts are not always systematically defined. Me and you might use the word infinite and expect a common understanding but because (subtlety) we are talking about two different contexts we will be in violent disagreement. If God hits me over the head with a rock and i get mad but never look at the rock, and never stop to look at the rock, i might have different results then if i immediately look at the rock and discover it looks alot like gold. People misunderstand your comments on infinite because they don't realize its a very important subject to investigate, and classification in some text might make coming to a common conclusion more likely.

It is the concept of an actual completed infinity in the real world that I am arguing against. There are two basic classes of infinite objects:

- Potential Infinities. The limit concept in calculus could be regarded as an example
- Actual Infinities. As represented by infinite sets / transfinite numbers in set theory

Finitists (of which I am one) might object to the first usage, but always object the second usage. Strict finitists do not even allow for potential infinities. In fact some even hold the number system should be restricted to only those numbers that are comprehensible to the human mind. Classical finitists allow potential infinities and do not believe there is a ‘greatest number’. I am a classical finitist.

So for example, I do not object to future time potentially going on forever as that would be an example of a potential infinity.

But I would object to the idea of an actual completed infinite set existing in the real world. So a collection of objects whose number is not finite I would object to.

Hope this is clearer...

But I would object to the idea of an actual completed infinite set existing in the real world. So a collection of objects whose number is not finite I would object to. — Devans99

i understand. Alot of the stuff you write has to be copied into a text file like note pad and analyzed line by line. That last statement i agree with.

I tried to google for "constructive real numbers are not complete", or something similar. — Mephist

Ok herewith my response to your Part 2. I'm buried in mentions again. Working hard to catch up.

I think this is what you refer to by "constructive reals". Is it?
Can you give me a link where is written that they are not complete? — Mephist

I posted earlier a proof that the computable reals are not complete. That's indisputable.

But it turns out that you are right, the constructive reals are not quite the computable reals, and I found many points of inconsistency in the literature.

First:

"The concept of a real number used in constructive mathematics. In the wider sense it is a real number constructible with respect to some collection of constructive methods. The term "computable real number" has approximately the same meaning."

https://www.encyclopediaofmath.org/index.php/Constructive_real_number

Ok, the constructive reals are sort of like the computable reals. Then this:

"The main message of the notes is that computable mathematics is the realizability interpretation of constructive mathematics."

http://math.andrej.com/2005/08/23/realizability-as-the-connection-between-computable-and-constructive-mathematics/

Ok, the computable reals are the "realizability interpretation" of the constructive reals. I'm not familiar with that technical phrase but it seems to indicate once again that the computable reals aren't too far off from the constructive reals.

But then there's this link you gave me:

https://users.dimi.uniud.it/~pietro.digianantonio/papers/copy_pdf/RealsAxioms.pdf

That paper (badly written and/or badly translated, and confusing) give a completeness axiom that they claim makes the constructive reals complete. But if that's true, then this is nothing more than an alternative axiomitization of the standard reals. Good for Coq, fine; but devoid of philosophical interest since all complete totally ordered fields are isomorphic to the standard reals.

By this paper, which you asked me to look at, the constructive reals ARE the standard reals and contain uncountaby many noncomputable reals.

So I admit I have no idea what the constructive reals are and I can't find two explanations that are consistent with each other.

Perhaps the problem is with the underlying murkiness of intuitionism. Or perhaps I haven't seen the right article.

But I'll conceded that evidently the constructive reals are different than the computable reals; but how, I can't say and don't know.

I am convinced that my definition of "constructivism" is not the same thing that your definition. — Mephist

After a day's research, I agree. I have no idea what the constructive reals are.

Well, here's a simple definition of what I mean by constructive logic:
=== A logic is called constructive if every time that you write "exists t" it means that you can compute the value of t. === — Mephist

Well than that is inconsistent with the article you asked me to read. Because that article's constructive reals are Cauchy-complete, and therefore must include many noncomputable real numbers. So your definition is not consistent with the definition in that article.

But if YOU are falling back on the requirement that constructive reals must be computable, then I already proved (twice) that they are not Cauchy complete.

I believe that you can define real numbers that are complete in a constructive logic. I think the example that I gave you using Coq is one of these. But I could be wrong: I am not completely sure about this. — Mephist

Yes, that's the problem. The link you gave me did do this. But then the constructive reals in that article must (a) contain lots of noncomputable reals, and (b) must be isomorphic to the standard reals, depriving them of any philosophical interest regardless of how Coq-compatible they may be.

I googled this: "non archimedean fields are not complete" and the first link that come out is this one:
https://math.stackexchange.com/questions/17687/example-of-a-complete-non-archimedean-ordered-field — Mephist

Yes, I agree I was wrong about my claim. However the hyperreals do happen to fail to be Cauchy-complete, so my Goldilocks remark stands.

Probably, as they say, "The devil is in the detail". I read several times in the past about Abraham Robinson's hyperreal numbers, and I believe that I read somewhere that non archimedean fields are not complete. So I believe that, under appropriate assumptions, this is true. But why is this a problem? — Mephist

Why is not being Cauchy complete a problem? Because then they fail to satisfy the ancient intuition of the continuum. I thought I explained all that.

Hmmm... I understand what you mean:
- "constructive" reals are computable functions. Then there is a countable number of them — Mephist

Yes. But there's even another wrinkle. The computable real numbers are countable; but they are not effectively countable. That's because any enumeration of the noncomputable reals must be noncomputable itself. Else you'd solve the Halting problem. So a strict computabilist can claim that the computable reals are NOT countable, because there's no computable enumeration of them! I've seen this argument in print.

- standard reals are the set of all convergent successions of rationals then their cardinality is aleph-1
- nonstandard reals are much more than this (not sure about cardinality), since for each standard real there is an entire real line of non-standard ones. — Mephist

Not a real line, a "cloud." Let's please not go off talking about the hyperreals. I only used them to point out that of the three famous models of the reals, only the standard reals are Cauchy complete.

Well, here's how I see it:
- "constructive" reals (with my definition) can be put in one-to-one correspondence with standard reals, only with a different representation — Mephist

If that's true then they necessarily contain many noncomputable reals, contradicting your own definition.

(but I don't know a proof of this) and do not correspond to computable functions. It is true that if you can write "Exists x such that ... " then you can compute that x, But for the most part of real numbers x there is no corresponding formula to describe them (and this is exactly the same thing that happens for non constructive reals). — Mephist

You're already contradicting your own definition. The constructive reals, whatever they are, are quite murky.

- Robinson's nonstandard reals are more than the standard reals because you exclude induction principle as an axiom (so that "P(0)" and "P(n) -> P(n+1)" does not imply "forall n, P(n)"). But there are objects used in mathematics that are treated as if they were real numbers, but DO NOT have the right cardinality to be standard real numbers: for example the random variables used in statistics: — Mephist

At this point I have to decline to go off in yet another direction. I actually wish you would make a brief, simple, clear point that I can work with.

https://en.wikipedia.org/wiki/Random_variable . So, they are more similar to nonstandard reals. — Mephist

I'll pass on this remark but I don't agree with it.

- The real numbers of smooth infinitesimal analysis are less then standard real numbers, and even the set of functions from reals to reals is countable: basically, every function from reals to reals is continuous and expandable as a Fourier series. And there are infinitesimals.
What for such a strange thing? Well, for example, they correspond exactly to what is needed for the wave-functions and linear operators of quantum mechanics: there are as many functions as real numbers, and a real numbers correspond to experiments (then, there are a numerable quantity of "real" numbers). And what's more important, a wave function contains a definite quantity of information, that is preserved by the laws of quantum mechanics. — Mephist

Fine, whatever. I would prefer at this point to constrain the discussion, not widen it. I was greatly disheartened after reading your own link and finding out that their model of the constructive reals is isomorphic to the standard reals. That undercuts everything I know about the constructive reals.

So, from my point of view, there is not one "good" model of real numbers, at the same way as there is not one "good" model of geometric space. — Mephist

There's exactly one model (up to isomorphism) that is Cauchy complete; and that is the standard reals.

[ END OF PART TWO :-) ] — Mephist

Ok. Next up, parts 3 and 4, and a big stack of mentions in the politics forum, which I'm afraid to read.

But can you do us both a favor and write a short, clear, and consise thesis that we can discuss? Part 2 was terrible for me because it contradicted itself in so many ways, giving multiple characterizations of the constructive numbers, at least one of which is isomorphic to the standard reals modulo an awful translation from the original Italian.

Let's take as reference the most complete proof of the theorem that I was able to find: — Mephist

I have to defer this for now. You already said you're not interested in the proof of B-T and now you've written a lengthy post about it. The proof in ZFC is unassailable, there is no point in your trying to find fault with it. And no point in my trying to explain it, since you shook my confidence in our conversation by claiming a translation in Euclidean space can't be continuous. I wish you'd keep your posts simple.

Right now my plan is:

* Catch up to parts 3 and 4 of your earlier replies;

* Catch up my mentions on the political threads;

* Try to get you to state a simple thesis so that I know what you're talking about. As it is you have given me definition of constructive reals that contradicts itself, along with a badly translated paper that defines the constructive reals as isomorphic to the standard reals. I can not keep up with all this as much as I'd like to.

* Urge you to split out a SEPARATE THREAD on Banach-Tarski. And start with a study of the paradoxical decomposition of the free group on two letters. That's the heart of the paradox and it doesn't require any set theory. It would be true in anyone's universe.

OK, this is the end. I think I cannot explain better than this my argument about BT. — Mephist

There's little point in your trying to find fault with an 80 or so year old theorem that's been checked and rechecked. But I would love for you to split out your B-T concerns into a separate thread so we can focus on them. Your claim that isometries aren't continuous or don't preserve topology is just terribly wrong, I honestly don't follow your reasoning at all. You also confuse measure 0 with nonmeasurable.

By the way I truly commend you for diving into the proof. You seem to understand most of it. That's impressive. I'm not sure what you mean about your objection but I'll try to understand it.

I think in that in an ideal mathematical language, Chaitin's Omega wouldn't be stateable. To say 'Omega is definable but non-computable' is surely not a statement about a number, but a statement about the syntactical inadequacy of our mathematical language for permitting the expression of Omega. — sime

You'll have to take that up with Chaitin and the mathematical logicians. As it happens, the class of real numbers definable in first-order logic is slightly larger than the class of real numbers computable by Turing machines. This is indisputable. I have no idea what an ideal mathematical language is. But in fact I only used Chaitin's Omega because it's a noncomputable real that we can visualize. The proof would go through with any noncomputable number. The n-th trunctions of any noncomputable number form a Cauchy sequence of computable numbers that fail to converge to a computable number. proving that the computable reals are not Cauchy-complete.

There are two basic classes of infinite objects:

- Potential Infinities. The limit concept in calculus could be regarded as an example
- Actual Infinities. As represented by infinite sets / transfinite numbers in set theory — Devans99

Assuming this is correct, which given your penchant for error and misstatement is going some. But given this, let's just consider the stronger which I assume will carry the lesser, actual infinite sets. If there are actual infinite sets then it would seem to follow a fortiori that there wold be potentially infinite sets.

Question: how many numbers are there? It's a fair question and a meaningful question. Answer!

Does the answer mean that you could produce a bean for every number and thus have a very large pile of beans that extended out of sight? Of course not! But the concept for the purpose is complete.and actual. QED.

Yes, and this clarifies a lot o things about infinity: — Mephist

Ok, herewith my reply to Part 3.

"In first-order logic, only theories with a finite model can be categorical." (form https://en.wikipedia.org/wiki/Categorical_theory). ZFC is a first-order theory and it has no finite model (obviously), than it cannot be categorical. Ergo, you cannot use ZFC to decide the cardinality of real numbers: "if a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities." (from the same page of wikipedia). — Mephist

Ok fine. But now you're spinning off into new topics. I can't keep up with this. I've learned a lot from this thread but once again I urge you to focus your concerns so that I can address them. Who the hell is arguing that first-order logic can determine the cardinlity of the reals? Who is claiming that????

Then, you can say, the problem is in the language: let's use a second or higher order language, and you can discover the "real" cardinality of real numbers. — Mephist

But I haven't said that!!!! You are arguing with strawmen. We already know that ZFC does not determine the cardinality of the reals. Who the hell is saying any different?

Well, in my opinion this is only a way of "hiding" the problem: it is true that if you assume the induction principle as part of the rules of logic, you get a limit on the cardinality of possible models (the induction principle quantifies over all propositions, so it's not expressible in first-order logic), but this is exactly the same thing as adding an axiom (in second order logic) and not assuming the induction principle as a rule. Ultimately, the problem is that the induction principle is not provable by using a finite (recursively computable) model: it's not "physically" provable. — Mephist

I have no idea what you are talking about. Honestly. I can't relate this to anything we've been discussing.

That's exactly the same situation as for the parallels postulate in euclidean geometry: you cannot prove it with a physical geometric construction (finite model), because it speaks about something that happens at the infinite, and the fact of being true or not depends on the physical model that you use: — Mephist

Once again I fail to relate this to anything we've been discussing.

Physical model? Physics has nothing at all to do with the independence of the parallel postulate.

if computers that have an illimited amount of memory do not exist, or, equivalently, if infinite topological structures do not exist, then the induction principle is false, and infinitesimals are real! — Mephist

My friend @Mephist, that is word salad and nonsense. We've come this far, but you're losing me. What you said is just false, to the extent that it has any meaning at all.

So, the sentence "if you use uncomputable (non constructive) axioms in logic, you can decide the cardinality of real numbers", for me it sounds like "if you use euclidean geometry, you can prove the parallel postulate". — Mephist

Well at least Part 3 was easy. It's incoherent. I don't mean to be provocative. Only that your other posts to me have been insightful and interesting; and Part 3 was nonsense.

But now I'm ready for Part 4 and my catching up to your posts will be a great accomplishment for me. I urge you, I beg you, to try to be concise. You're burying your own points in words.

HOTT is not a constructivist theory (with my definition of constructivism) because it uses a non computable axiom: the univalence axiom — Mephist

I didn't know that. But ok. If HOTT is not constructive, what are we talking about?

And now after all this: What ARE we talking about? I no longer know what we're discussing. You lost me somewhere and you've refused to throw me a lifeline.

It's been interesting as hell, but the last two of your 4-part post really lost me.

This was considered by Voevodsky as the main "problem" of the theory, and there are currently several attempts to buid a constructive version of HOTT. One of them is cubical type theory (https://ncatlab.org/nlab/show/cubical+type+theory), but I don't know anything about it. — Mephist

Ok that's very interesting. Poor Voevodsky.

[ THIS WAS THE LAST PART :-) ] — Mephist

Yay! I made it!!!! Not sure what I accomplished, but at least I got caught up in this thread.

Can you show me a physical theory, or a result of a physical theory, that is somehow derived from the fact that a continuous line is made of an uncountable set of points? — Mephist

You mean besides relativity and quantum physics? The 't' in the Schrödinger equation is a continuous parameter over the real numbers. What math do you think the physicists are using?

Of course I'm not saying that the physicists MUST use continuous math; only that to date, they do.

Formal logic (currently assumed as the foundation of mathematics) is only dependent on one very fundamental fact of physics (that usually is not regarded as physics at all): the fact that it's possible to build experiments that give the same result every time they are performed with the same initial conditions. — Mephist

Wow. I don't know where to start. That couldn't be more false.

Mathematics (what is called mathematics today) is the research of "models' factorizations" that are able to compress the information content of other models (physical or purely logical ones). A formal proof makes only use of the computational (or topological) part of the model. The part that remains not expressed in formal logic is usually expressed in words, and is often related to less fundamental parts of physics, such as, for example, the geometry of space. — Mephist

Not sure what you're getting at, but nothing I relate to.

Riemann understood that the concepts of "straight line", measure, and the topological structure of space are not derivable from logic, but should be considered as parts of physics. — Mephist

His work on non-Euclidean geometry was purely mathematical. But if you understand that Riemann knew that math wasn't physics; why won't you make that same distinction?

In te future, when mathematicians will start to use quantum computers to perform calculations, — Mephist

Mathematicians don't use computers to do calculations. This is a very naive point of view of what mathematicians do. It's often been remarked that of all the STEM fields, math departments don't use computers! The proof of the four-color theorem was an anomaly in 1976 and remains an anomaly today.

I believe that even the existence of repeatable experiments will not be considered "a priori", but as an even more fundamental part of physics. — Mephist

You've persistently claimed that mathematicians are doing physics, but this is an engineer's view of math. Mathematicians don't do physics. Just ask the physicists!

So, there will be quantum logic that is more powerful than standard (or even constructionistic) logic, at the price of not being able to be 100% sure that a proof is correct (but you will be able, for example, to say that we are sure about this theorem with 99% of probability). — Mephist

It's true that there are probabilistic proofs, but those are a subset of math, not all of math. Not even most of math. Mostly, a little bit of math.

Surely your ( and most of other peoples' ) reply to what I just said is that "this is no more mathematics". — Mephist

Not the first time you've put words in my mouth that I didn't say, wouldn't say, and don't agree with. But nobody would disagree that probabalistic proofs exist. But even the guy who wrote the famous Death of Proof article in Scientific American had to backpedal.

Well, at the time of Euler topology was not mathematics either. — Mephist

You seem to be grinding an ax but I'm not sure about what. That future math will be different than current math which is different than past math? Ok. Why would anyone disagree?

I hope going forward that you'll write less but with more focus. You really wore me out and in the end you yourself agree that you're not sure what the constructive reals are. As Gauss said: Few, but ripe.

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There's exactly one model (up to isomorphism) that is Cauchy complete; and that is the standard reals. — fishfry

That seems to be the common point of all your arguments about real numbers, so I wanted you to show you this: https://mathoverflow.net/questions/128569/a-model-where-dedekind-reals-and-cauchy-reals-are-different
I know that there is a proof of uniqueness.

That seems to be the common point of all your arguments about real numbers, so I wanted you to show you this: — Mephist

Yes but first, that in no way invalidates the fact that Cauchy-completeness uniquely characterizes the standard real numbers up to isomorphism; and secondly, that the comments and answers in that thread are "inside baseball" remarks from professional specialists in the field. I can understand the words but not the meaning. I have to assume the same is true for you, unless I'm wrong. I'm aware that Cauchy completeness is not the same as Dedekind completeness.

Example from one of the comments: "You won't find a model of ZF where they are different but there are models of IZF where they are different."

Now this is not something I'm prepared to say I understand even if I can parse the words. I assume the same must be true for you. This thread is very inside baseball, it's not for amateurs like us. IZF is evidently intuitionistic ZF. More mysteries beyond my pay grade.

But gee, I only meant to put in a good word for the standard reals. I'm not driving a stake in the ground and vowing to defend them to the death. Everyone knows they're murky.

The one thing I do put a stake in the ground about is that there aren't enough Turing machines to plug the holes in the computable reals. But apparently this is less clear when it comes to the constructive reals, which at least that one Italian paper seems to think are Cauchy complete. But how can that be? Apparently there are constructive approaches to Cauchy completeness. I confess great bafflement in this regard.

Here is a good explanation of what "contructive mathematic" means: https://www.iep.utm.edu/con-math/

Here is a good explanation of what "contructive mathematic" means: https://www.iep.utm.edu/con-math/ — Mephist

Thanks man I'll get to this later. I am most definitely worn out for the day. What I did learn from all this is that the computable reals are NOT the constructive reals; and that the constructive reals can be Cauchy-complete. Now THAT I did not know, and it causes me to confront the depths of my ignorance. I'm really mystified on this point.

One article I'll definitely be rereading is Andrej Bauer's Five Stages of Accepting Constructive Mathematics.

http://math.andrej.com/2016/10/10/five-stages-of-accepting-constructive-mathematics/

Yes, that's very interesting. Thank you for the reference!

P.S. After reading this, please read my explanation on what is "wrong" with Banach-Tarski paradox, if you have time.

Would 1/0 = a vacuum/blackhole? -abstract thoughts

Would 1/0 = a vacuum/blackhole? -abstract thoughts — Pomme

No. Math is not physics. That seems to be a theme today.

In fact whenever mathematicians run into infinities, they do NOT say Aha here's an actual infinity in the real world. Some amateurs do that but professional physicists never do.

Rather, they say: "Our model has broken down. We don't know what happens below a certain scale, which is small but greater than zero."

There's an elementary example that most people should be familiar with. Consider classical Newtonian gravity,

$F = G \frac{m_1 m_2}{r^2}$

where $F$ is the gravitational force, $G$ is the gravitational constant (just some number that makes the units work out), $m_1$ and $m_2$ are the masses of the two bodies, and $r$ is the distance between their centers of mass.

Newton proved a theorem that if the bodies are spherical, you can just use the center of masses and it's the same as if you added up the gravitational attraction between all the respective pairs of points in the two spheres, a much harder calculation. Basically Newton did multivariable calculus before there was multivariable calculus.

Because of this theorem, it's common to think of massive bodies as point masses; that is, dimensionless points that nevertheless have mass.

Now if you have two point masses, they can get arbitrarily close together. As the distance between the two point masses gets close to zero, the gravitational energy between them gets arbitrarily large. It "goes to infinity" as they say.

Physicists do NOT say, "Wow a black hole with infinite energy results from two point masses getting too close together." We might think of it that way in a late night session at the dorm under the influence. But in the cool light of day we must say: "Newtonian gravity breaks down below a certain distance scale. When two sufficiently small masses are very close together, the gravitational energy is greater than all the energy in the universe. The equation may not be applied below a certain scale of distance.

You can see that even without quantum theory, we can show that there's a minimum distance, below which the theory breaks down. There's a sort of Planck length even in Newtonian gravity.

Assuming this is correct, which given your penchant for error and misstatement is going some. But given this, let's just consider the stronger which I assume will carry the lesser, actual infinite sets. If there are actual infinite sets then it would seem to follow a fortiori that there wold be potentially infinite sets.

Question: how many numbers are there? It's a fair question and a meaningful question. Answer!

Does the answer mean that you could produce a bean for every number and thus have a very large pile of beans that extended out of sight? Of course not! But the concept for the purpose is complete.and actual. QED. — tim wood

Numbers exist in our mind only. Actual infinity can exist in our minds only along with other things that are not logical / do not exist in reality like square circles. But just because you can think of actual infinity does not mean you have realised/completed actual infinity in your mind.

How many numbers are there?

- First thoughts: there are a potential infinity of numbers. We can go on counting forever so its a potential infinity. But of course we would lose track after a while. So this potential infinity has a finite, mind related limit.

- Second thoughts: in no way is the human mind capable of realising an actual infinity of numbers - the mind in finite after all - there is a limit on the largest number than can be conceptualised - the number of bits representing the number must fit into the storage capacity of the human mind. A very large number then, and dependent on the mind that's conceptualising it.

So actual infinity is not realisable in our minds or in reality.

So actual infinity is not realisable in our minds or in reality — Devans99

Are we the standard by which we measure infinity? Isn't this like saying tortoises that live upto 300 years don't exist because humans only live upto 100 years?

Are we the standard by which we measure infinity? Isn't this like saying tortoises that live upto 300 years don't exist because humans only live upto 100 years? — TheMadFool

Numbers exist in the mind only so I guess you are correct - we should not limit our consideration to the human mind. An alien with a larger mind would be able to mentally realise larger numbers than a human. But they would still be limited to conceptualising finite numbers as per the storage capacity of their minds.

Minds exist in reality so all minds are a finite collection of storage elements (I hope we agree that realising an actually infinite collection in reality is impossible). So whatever the actual size of the mind, the largest possible number is still finite.

I remember you mentioning the Grandi series

Grandi series: $1-1+1-1+1-1+...$

It's used to prove that $1+2+3+4+...= -1/12$.


Does the above have anything to do with your views on infinity?

$\frac{-1}{12}$

With Grandi's series, at any finite point in the series, its sum is either 1 or 0. So the sum of the infinite series (if it has a sum) must be 1 or 0. But mathematics cannot be sensibly evaluate Grandi's series - standard methods produce 1/2 or contradictory results.

A more physical demonstration of Grandi's series is Thompson's lamp - the lamp is on for the 1st second, off for the next 1/2 a second, on for the next 1/4 of a second, and so on. After 2 seconds, is the lamp on or off? It must be one or the other, it cannot be 1/2 on.

Both problems are examples of 'supertasks'. In both cases, knowledge of whether actual infinity is odd or even is required, which requires knowledge of the size of the associated sequence representing the series. But an actually infinite collection of objects is an impossibility - so the sequence cannot be said to have a size - so we cannot evaluate it.

I think these are illustrations that all supertasks are impossible - anything with an actually infinite number of steps cannot happen in finite time. In the case of the lamp, something physical in reality would limit the number of steps in the supertask to a finite number - probably time is discrete - planck time for example - then the series could not go on forever and the lamp would definitely be on or off at the end of 2 seconds. Discreteness addresses Zeno's paradoxes too (which are also examples of supertasks).

I think the fact that maths cannot evaluate the series is supportive of the view that actual infinity does not exist as a (sound) mathematical concept and does not exist in reality.