But you’d never finish counting the reals between 0 and 1 so you can’t completely define the set. — Devans99

In math, counting in understood rigorously, e.g. one-to-one correspondence. I'm not talking about the temporal process that people do.

And no way is the set bounded in terms of precision; that stretches to infinity so it’s unbounded. — Devans99

That has absolutely nothing to do with being bounded or not. There are numbers greater than all those in the referenced set, and numbers lesser than them. That's a bound, you're grasping at straws my dude.

But my main point you ignore - numbers have size zero so they do not exist - so talking about how many you can get on a number line between 0 and 1 is nonsense. — Devans99

I really don't think you understand the purpose of a number line.

The Reals between 0 and 1 are unbounded in terms of precision. Imagine writing out all such reals to 1 decimal place (0.1, 0.2, etc...), then to 2 decimal places, then 3 etc... This is an example of potential infinity. — Devans99

Utterly irrelevant. They have a specific magnitude, they are demonstrably greater than 0 and lesser than 1. The length of their decimal expansion has no bearing on the boundedness of the set. It's bounded. It's not a potential infinity, it's an actual, literal infinity.

When you’re working out how many things compose another you take the overall length and divide by length of the constituent parts. So to work out how many points there are in the interval 0,1 you divide 1 by point size.


The problem with the number line example is that numbers have no length. They are labels that have no length. They don’t exist. So the number of numbers between 0 and 1 is 1 / 0 = undefined which is what you’d expect.

What? You just count them, and counting is well understood mathematically. The set of numbers between 0 and 1 has the cardinality of the continuum. It's clearly bounded, there are numbers larger and smaller than any element in the set.

Yeah I mean constructivists have issues with Choice, but constructivism has a lot going for it, and it's been around about a century now so that's not surprising. It really jives with computation and Choice isn't always needed, so there's some rationale for not always using it (though on pace, I think Choice is perfectly fine so I wouldn't be all that attracted to intuitionism). We often have a choice (zing!) with using Choice, but dropping the infinity axiom is needlessly limiting to math.

I am not inclined to drop the idea that the natural numbers are infinite, only the idea that the infinite natural numbers are a set. — Metaphysician Undercover

Good luck doing that without the rigorous mathematical understanding of infinity as opposed to the vague colloquial understanding.

As andrewk indicates, if your axiom states that a set may be finite or infinite, then that is what is the case in that axiomatic system. The problem that I see, is that the way "set" is used by mathematicians, as a closed, bounded object, the possibility of an infinite set is precluded. Sets are manipulated by mathematicians, as bounded objects, but an infinite set is not bounded like an object, and therefore cannot be manipulated like an object. This calls into question the understanding of "infinite" which is demonstrated by this axiom of infinity, which stipulates that the infinite natural numbers are a "set".

Infinite sets can very well be bounded, I've already given an example. The set of reals between 0 and 1 is provably infinite, and clearly bounded. After all, every element in that infinite set is larger than 0 and yet smaller than 1; they literally are between bounds. But whether or not sets are bounded or not really has nothing to do with infinity. A set whose members are ever increasing due to some iterative calculation is clearly unbounded, but it's not infinite. Just loop a program which adds new members to an array every iteration; at every iteration the number of members of the array are obviously going to be finite.

OK, but the question was, how does your giving a rule populate a set? Do you apprehend the issue. Suppose I decree, as you suggest, that all red things are members of a set, the set of red things. How does this declaration make certain things members of that set, while excluding other things? — Metaphysician Undercover

A set is defined by said rule applying to the objects in question. It has nothing to do with the process of collecting things. The properties in question are possessed by those objects whether or not I accept they do or if I call it something else. Being in the set of African Americans doesn't depend on anything to do what I call them. It doesn't necessitate an essential meaning, just a conventional one which people roughly agree picks out a certain class of objects.

You think that it is simple to name some condition which applies to all of a number of objects? This could only work if the condition which is named had an essential meaning, an official definition, allowing that all the things could be judged according to that definition.

I've just answered this. It has nothing to do with judgements. This is akin to saying that because I haven't looked at all the even numbers I can't declare them to be part of the set of even numbers. So long as they fall under the same rule or property specification then my directly "collecting" them is entirely optional, and really irrelevant.

Stipulating that certain numbers fall into a certain set does not make that so. — Metaphysician Undercover

I gave a rule that populates members of a set, I do not literally gather abstract objects and place them somewhere.

But the original point I made, on the other thread, is that you cannot stipulate the existence of a set, because "set" is defined by "collection", and collection requires collecting. Do you see the circularity of your begging the question? — Metaphysician Undercover

You're using a colloquial usage of collection, and not even the only colloquial use of that words. People have spoken of the collection of stars in the sky, only a child would think they literally meant they gathered the stars into the sky as opposed to a condition that applies to some class of objects

So I see a number of objects, and I stipulate, those objects are a collection. What makes them a real collection rather than just an imaginary collection? — Metaphysician Undercover

Name some condition which applies to all of them or just create an extensional list of the objects. It's seriously simple.

If it is not the act of collecting them into a group, and demonstrating that they have been collected, then what is it? Would you argue that sharing essential properties is what makes them a collection? Who would determine which properties are essential, and which are not? — Metaphysician Undercover

It's not essential properties, it's just whatever properties you declare the set to be constructed based on. The "set of all red things" is, quite obviously, populated by all the objects that have the property of being red.

As you point out, we lose quite a lot of mathematics by dropping the Axiom of Infinity. And to me the foregoing arguments for doing so are ridiculous since the claim is contradictions occur with the axiom (they don't). Even Ultrafinitists don't claim that provable contradictions appear, so the justification for dropping the Axiom just looks like philosophical bias more than anything.

I agree.

Nope~

That's not the debate I was asking for in the OP. It was whether the words themselves have inherent meaning that straying from is necessarily changing the subject in a ridiculous way.

Exactly, this is what the quotation is saying, "infinite" in calculus and algebra is different from "infinite" in set theory. Set theory has transfinite numbers, alephs, but the definition of "infinite" in calculus and algebra is defined in relation to limits. — Metaphysician Undercover

This is what I'm talking about. "Infinity" in the context of limits might mean something else (emphasis on "might"), but calculus still uses multiple levels of infinity as understood in set theory, because we understand calculus through set theory. Hell, even in limits I could just assume the infinit there refers to Aleph-null and the calculation is still going to work. All it needs to mean is that it's larger than whatever I'm working with. And Aleph-null is necessarily larger than any finite number. Wikipedia is a poor source.

The point being that there is no clear definition of "infinite" in mathematics as you claim, the definition varies. In geometry for example, a line is endless, infinite. Contrary to your claim, "boundless" is a valid definition of "infinite". — Metaphysician Undercover

In geometry, lines are continuums which are captured in set theory. There is a clear definition of infinity(ies), you just don't seem to get that meaning here is context sensitive. "Boundless" as a definition of infinity is patently stupid because plenty of infinities are bounded. The set of real numbers between 0 and 1 are indisputably infinite, and yet it is bounded between 0 and 1. That's just an obvious reason why that definition won't work. Colloquial definitions are inherently vague and only make sense in certain contexts. Real mathematics is not one of those places these sloppy definitions will work in.

So your defence, which was nothing more than an appeal to authority is lame and vacuous. Because the various mathematical authorities have various ways of defining the term, we cannot trust that any of them really knows what "infinite" means. — Metaphysician Undercover

I'm sorry, this is not only a misrepresentation of what was said before but is thoroughly ridiculous given it requires pretending words don't intentionally change meaning in the appropriate context. Waste of time.

Ehh, I think the issue there is what the brain does cognitively when we think about words is that we've already internalized "this means that because that's how people in my life have used it to mean". Nothing about "red" inherently makes the mind conjure up a particular range of colors, just ask a pacific islander who doesn't speak a lick of English.

Until you demonstrate that "set of natural numbers" is not self-contradictory, such claims are nonsense. And to say that something infinite is not indefinite clearly is contradictory. So carry on with the nonsense. — Metaphysician Undercover

Infinite sets are not indefinite, why do you keep saying this as if it's an obvious fact that I've conceded? Every object that's a natural number will fall into that set once I've stipulated an intensional definition of that set. You haven't once shown it to be contradictory, you just fall back on saying that anytime you're challenged to defend your position. The set of numbers equal to or greater than zero is a perfect consistent, definitely set. If you don't understand what the members of that set are, then that's because you don't understand the definition.

Well, I'm not sure that's quite the case under intuitionism, where infinity is only a potential, and the only natural numbers that exist are the ones which have been stated, written down or computed. — Marchesk

Sure but I'm assuming that none of the other users here are doing constructive mathematics. In classical, standard mathematics, what I said is completely true. I'm all for discussing alternative maths and logics (I'm a logical pluralist), but I ignored it for simplicity.

However, as a matter of fact most constructivists nowadays do accept that the set of natural numbers (and any other countably infinite set) is actually infinite.

This is why you don't quote Wikipedia, especially when it's not a topic you're familiar with. The infinity referred to there is not a number. Limits do not diverge to a number per se (or if it does, it's to some transfinite number), they just increase without bound which meets a colloquial meaning of "infinity". But the transfinite cardinals and transfinites ordinals are indisputably numbers. Infinity is well understood in mathematics. Infinite sets have a cardinal number which is infinite, but that has nothing to do with what's referred to in limits.

So yes, I think this has run its course for me. Infinity in that other sense essentially means "larger than anything else you have", not necessarily a specific number.

That's fine, but if there's an infinite amount of such an element, I don't see how this qualifies as " "quantity". Don't you know that "quantity" is defined as a measurable property of something, or the number of something" Infinite is neither a measurable property nor is it a number, so you really haven't given me a definition which allows for infinity. — Metaphysician Undercover

The number of natural numbers is the infinite cardinal aleph-null. Ergo, by your definition it's a quantity. QED.

Really, I wish you would give more thought to what you say MindForged. How could "infinite" signify a quantity? Any such so-called "quantity" would clearly be indefinite and therefore not a quantity at all.

"How could it?" How could it not? It's not indefinite, the members of the "set of natural numbers" never increases or decreases, it is exactly what it is and has always been.

If you knew the precise cardinality of an infinite set, you'd be able to tell me the relationship between the cardinality of a finite set and that of an infinite set. Obviously you know of no such relationship, as subtracting a finite number from an infinite set does not change its cardinality. There is no such relationship. Therefore my suspicions are confirmed, you really do not know the cardinality of an infinite set. Your claim was a hoax. And so your assertion that "infinite set" is not contradictory is just a big hoax. — Metaphysician Undercover

You aren't making sense. I just told you the difference. I already walked your through the informal proof, but not once have you actually acknowledged it. If I take the cardinality number aleph-null, the the size of the natural numbers, and remove the element that's the number Zero, the cardinality doesn't change, e.g.

1 - 0
2 - 2
3 - 4
etc

And so subtracting a finite number of elements from an infinite set won't change the cardinality. What relationship are you looking for? Any finite number will be.lesser than the cardinality of any infinite set, so subtraction here won't do anything.

Seriously, you either are terrible at making your point or you are more concerned about this at an ideological level and reflexive reactions than one of philosophy.

I know you feel this way, that's why I've proceeded to, and succeeded in demonstrating that "infinite set" is contradictory according to your definition, and the one used by mathematicians. Clearly an "infinite set" is not a well-defined collection in any mathematical sense, because the cardinality of such a set is not at all well-defined. Therefore it cannot be a well-defined collection, mathematically, and cannot be a mathematical "set". — Metaphysician Undercover

You did no such thing. You claimed it was ill defined. I showed the informal proof of it being an infinite set (the one-to-one correspondence argument) and you couldn't even address it. There is no known contradiction related to these infinities in modern mathematics. Anyone claiming there "clearly are" contradictions just don't know anything. Goodbye.

You should have specified what you meant by difference. I assumed you were asking how such sets were any different than a purportedly infinite set, so I gave the difference. If you were talking about the difference as in subtraction, then the answer is infinity. If I subtract any finite number from an infinite number, it's not going to change the cardinality. It's only finite numbers whose cardinality decreases when removing finite numbers of elements. If I take the natural numbers and remove the element Zero, it can still be put into a one-to-one correspondence with the even numbers, so this just provably doesn't change the size of the set.

And as I said, I don't care if it's a set according to your definition. Mathematicians don't use your definitions of these terms. They use the ones they stipulate, so that's what I'm obviously going to go with.

OK, then I suggest you quit using "transfinite", because you are only introducing ambiguity. Why then did you say: "The cardinality of the set of natural numbers is the transfinite number aleph-null." If "transfinite" is just an artefact, and transfinites are really infinite, then infinite sets really have no distinct cardinality, they are simply "infinite". — Metaphysician Undercover

Because that's what the numbers are called. http://en.wikipedia.org/wiki/Transfinite_cardinal. Further, your comment that they have no "distinct" cardinality because they're infinite does not follow. The way cardinality is determined is exactly how we know the set of naturals have a cardinality of aleph-null.

Your claim was that an infinite set has a precise and known cardinality. If this is the case then you can show me the relationship between the cardinality of an infinite set, and those other two finite sets, and how the difference between the cardinality of the two finite sets is expressed in the two relationships between each finite set, and the infinite set. — Metaphysician Undercover

Man, didn't I just do this? I showed the cardinality of the naturals between 1 and 100 and how we determined that. I also showed previously that the same means of determining cardinality, when applied to the entire naturals, results in a proper subset of the set having the same cardinality as the parent set.

Look, I'll try again for completeness.

"A" = set of naturals between 1 & 100
"B" = set of naturals between 1 & 200

To determine which is larger, we will pair each number from beginning to end with exactly one number from the other set. One element from A mapped to one element from B (A's on the left, B's on the right)c

1 - 1
2 - 2
3 - 3
etc.

99 - 99
100 - 100

Now we've hit a problem. Set A has no more members, we can't pair anything else up with the members of set B. And the reason is perfectly transparent: Set B has a larger cardinality, it has more members. But note what happens if we take the natural numbers (everything 0 and greater) with the even numbers and try to pair them off this way:

0 - 0
1 - 2
2 - 4
3 - 6
Etc.

Neither set ever fails to have members to pair off. That cardinality is infinity. The comparison to the other sets you mentioned, as I've said several times now, is that the naturals cannot be paired off with sets like the one you gave. Because sets like the natural numbers can be put into a one-to-one correspondence with a proper subset of themselves, they cannot be put into such a correspondence with sets, like those you gave, which cannot be put into that correspondence with a proper subset of themselves. Only infinite sets can do this.

OK, now we're getting somewhere. You were not talking about "infinite", or "infinity", you were talking about transfinite numbers. Why didn't you say so in the first place? This thread appears to be concerned with the "actually infinite". Transfinite numbers are something completely different, and I guess that's what caused the confusion, you did not properly differentiate between these two, nor did you let me know that you were talking about transfinite numbers rather than infinity. — Metaphysician Undercover

OP has been arguing against the coherence of infinity, including infinite sets. Qlso, I have repeatedly mentioned the transfinite numbers. I am talking about infinity, transfinite numbers are infinite. A set whose members can be put into a one-to-one correspondence with a proper subset of themselves (like the naturals) are infinite. "Transfinite" is more of an artefact in mathematical language from times where there was some dispute about the numbers, no mathematician nowadays thinks such numbers are anything but infinite.

Wait, now you're claiming that this demonstration which you produced earlier shows that a transfinite number is infinite. Care to explain, because I really do not see any demonstration of that. — Metaphysician Undercover

I've just explained this. Transfinite numbers are infinite. They meet Dedekind's definition of infinity, don't be confused by the name "transfinite". Finite sets can't be put into a one-to-one correspondence with a proper subset of themselves, as you'll end up with members in one of the sets running out because finite sets cannot have part of the set be the same cardinality as the entire set.

Come on, give me a break. If you're not joking about this, then how gullible do you think I am? If you actually believe that you know the exact cardinality of the set of natural numbers, then show me the precise relationship between the cardinality of the following sets. The set of natural numbers between 1 and 100, the set of all natural numbers, and the set of natural numbers between 1 and 200, — Metaphysician Undercover

I'm somewhat confused about the relevance to infinite sets. The set of natural numbers between 1 and 100 (call it "A") has a cardinality of 100. The set of natural numbers between 1 and 200 (call it "B") has a cardinality of 200. Set A cannot be put into a one-to-one correspondence with B since the cardinality of B is greater than that of A.

Neither A nor B can be put into a function with a proper subset of themselves (again, any subset will run out of numbers to pair with the parent set) and are therefore finite; try to match up 100 things with 200 things and you'll be able to see that's it's impossible to pair up one thing in one set with exactly one thing in the other set for all the members. This is exactly the difference between finite and infinite sets. Infinite sets can have parts of the set have the same cardinality as the entire set because you never can "run out" of members to pair up. That was the point of my earlier example with the Natural numbers and the Even numbers.

Well one-one correspondence is logically flawed: There are the same number of natural numbers as square numbers? Surely a paradox - a sign we are dealing with a logically flawed concept. — Devans99

How is it "surely a paradox"? That they can be put into a one-to-one correspondence shows they are the same size.

They were one of the two users, yes. :3

Can you help me, by describing what a collection would actually be, if we allow that collections may be infinite. — Metaphysician Undercover

An arbitrary quantity of elements referred to as a whole and which gain membership in said whole by means of sharing a common property we pick out or by being subject to the same stipulated rule.

E.g. the collection (set) of African Americans. Membership in that set is gained by the usual means and it wouldn't make a difference if there were ten of them, ten million of them or an infinite number of them. The above definition in no way can be said to require the number of members to be finite.

Actually, I define terms like "set" "collection", "object", and "infinite", in the ways normally accepted in philosophy. — Metaphysician Undercover

I forget, are mathematicians doing math or philosophy?

Worse, most philosophers who actually study maths too will employ the mathematical definitions of these. It's part and parcel of just using standard mathematics and classical logic.

Again, you keep talking about "their very definition" and pretending you don't simply means "the definitions I happen to use". Words are defined by their users, they don't have free floating meanings so your entire approach is fundamentally ridiculous.

1) A "set" is a well defined collection.
2) A collection which has an unknown cardinality is not "well-defined", in any mathematical sense.
3) If a collection were infinite its cardinality would necessarily be unknown.
4) Therefore an infinite collection cannot be well defined in any mathematical sense, and cannot be a "set". — Metaphysician Undercover

#2 is just an outright misrepresentation. Infinite sets do not have an "unknown cardinality". The cardinality of the set of natural numbers is the transfinite number aleph-null. This is demonstrated by simply looking at the mathematical means of determining the cardinality of a set, namely when we known sets have the same size as other sets. Any set which can be put into a one-to-one correspondence with a proper subset (meaning sharing some of its members but not having all of them of itself) is what defines an infinite set. The natural numbers have this property. Take the even numbers (which are half the naturals) and you can pair them up with the naturals and never fail to establish a pair, e.g.

0 - 0
1 - 2
2 - 4
3 - 6
etc.

No finite sets can have this property, as eventually you'll find they run out of numbers to put in a function. And as Cantor showed using his diagonal argument, on pain of contradiction we know the set of natural numbers is a smaller infinity than the set of real numbers as the reals are uncountably infinite.

Unless you can argue that the notion of a one-to-one correspondence is logically incoherent, you have no recourse against these well established mathematical tools. The idea you tried to pass off earlier that arguments from authority are off the table was ridiculous. Arguments from authority are an informal fallacy, meaning they are only invalid in specific cases. Namely, when the source is not actually an authority on the subject. In this case, my authority is quite literally nearly the entirety of the mathematicians.

#3 is incorrect for the previously stated reason. We know the exact cardinality of the set of natural numbers, real numbers (etc.) And those are infinite sets by the mathematical definitions of these terms. #4 just falls out as false because the premises used to establish it were false.

I've explained to you how "infinite set" is clearly contradictory. Also it's quite obvious that the waythe concept of "imaginary numbers" treats the negative integers contradicts conventional mathematics. You can rationalize these contradictions all you want, trying to explain them away, but that is just a symptom of denial, it doesn't actually make the contradictions not contradictory. — Metaphysician Undercover

Um, no. Literally you're entire argument is that "collection" and "set" are necessarily finite because of the definition your use. Your argument is without any force because it's indisputable that mathematicians don't use your definitions of these terms. It's entirely besides the point to try and claim they're incorrect for doing so by the means you're doing it. It's like saying "marriage" is definitionally between men and women and so the idea of gay marriage is a contradiction.

If space is a thing, it's not the same as the natural understanding of a thing. That's what I was talking about.

You're other point wasn't what I was talking about. I'm not saying space is infinite in breadth, but it can be infinitely divided without hitting some kind of base unit or boundary point.

They don't produce two different concepts. The extensionally defined set {3,5} and the intensionally defined set "odd numbers which are greater than 1 but less than 7" define exactly the same set. It's the same concept, a well-defined collection. Bye~

It's not a label for two different concepts, it's two ways of defining instances of the same concept. Intensionally defined sets are not incomplete, and they are certainly not undefined. I think I really am pulling out of this now.

But a set is a list of elements, if you don’t list the elements you are missing out the definition of the set. — Devans99

A set is not [merely] a list. A list can contain members of a set. The set of real numbers is unlistable (uncountable), but it's still a set. Listing out the members is only one way to define a set.

When we say ‘the set of bananas’ we are not defining a set, just specifying the selection criteria for the set which is a different thing from the actual set.

For example the actual set of bananas has a cardinality so clearly the actual set definition contains more information than the selection criteria.

The selection criterion is used to define the set. That's what an intensionally defined set is. The whole point of such definitions is that extensionally listing things is often not possible to do when defining something, especially a set. I can't list all the even numbers, but I can intensionally define their set.

[Mods please delete this, it double posted]

Actually no. Cantor's set theory is totally rigorous and logical. It doesn't fall into the paradoxes — ssu

Cantor's set theory did fall into numerous paradoxes because of the naive comprehension scheme. It was, as you said, ZF that avoided them using the separation and foundation axioms.

What is sound about the ‘set of all sets does not exist’? It exists as much ‘as the set of Naturals’ yet it does not exist in set theory.

But anyway, neither of the above are fully defined sets. You have to list all the members to fully define a set. — Devans99

What is a non-existent set? If you've defined it, it exists. So that's just the assertion of a contradiction.

You don't have to list out all the members of a set to define it. Seriously, sets are defined intensionally all the time.

Honestly, I don't think mathematicians care about contradiction within they're work — Metaphysician Undercover

I can say that as someone who has studied dialetheism and paraconsistent logic, and has mentioned such to things to friends of mine doing their grad degree in math (or just taking higher maths courses) that this is flatly untrue. If there are any actual, provable contradictions in standard mathematics, the law of explosion entails every sentence becomes a theorem. This is obviously not a good conclusion to draw in normal mathematics, just look at Russell's Paradox before we had ZF set theory.

Really, there's no evidence any of standard mathematics entails a contradiction, provided you actually use the definitions mathematicians actually use.

I don't see why this follows.

http://mathworld.wolfram.com/Collection.html

Are you using "collection" in this specific sense? But wouldn't that mean that a collection is a (countable) set, rather than a set being a collection? — Banno

No no, I'm saying that in the thread we could at least agree that sets are a type of collection (the everyday meaning of the words "collection"). The disagreement was whether or not "collection" necessarily implies a finite group of objects. In the bit you quoted I'm just granting that for the sake of argument, it's not about multisets.

Yeah, that was before we realized the naive comprehension scheme resulted in Russell's Paradox and trivializes the math if you keep the classical logic. There is no "set of all ordinal numbers" so Cantor's notion of an "absolute infinity" cannot be expressed in ZF set theory.

Sex with sex dolls

Starting with the natural numbers, every time we enlarge the set of numbers, the algebraic properties change. There's no reason for us to be surprised when it changes yet again when we move from the reals to the cardinals (including transfinite cardinals). — andrewk

Yeah that's why I didn't see it as a relevant objection even if it's technically correct. Your points were well made though. *thumbs up*

I don't see how an instantiated infinity could ever be established empirically since we can't count to infinity. On the other hand, I think in some cases, infinity can be ruled out. — Relativist

Sure but that's not really how one gets to infinity in math. It's not like when someone says the natural numbers are infinite they mean they've counted to some point called infinity. As you know, in modern mathematics it means the set can be put into a one-to-one correspondence with a proper subset of itself. If making the assumption that something in the universe (space, time, something else) is infinite is a part of a very good theory, that's perfectly reasonable a basis to think reality is infinite in that respect, even if in other respects it might not be possible. The Everettian/Many-Worlds Interpretation of QM seems really solid to a lot of physicists, and it seems to make such an assumption about the number of worlds, for example.

Edit: SophistiCat has already put it better than I:

If a model that makes use of infinities provides a good fit for many observations, is parsimonious, productive, fits in with other successful models, etc. then we consider it to be empirically established, infinities and all. — SophistiCat

Nah it's fine. It's technically not the exact same operation but since it behaves similarly enough I don't know what else to call it (additive operation?) My initial points were that infinity isn't inherently off the table when talking about reality, as the OP and another user were arguing that infinity is a contradictory concept (which is just flatly untrue); so if anything in reality is infinite or not is an empirical matter, there's no strictly logical argument against it being instantiated. Anyway, sorry if I was unclear!

It's not true that the "normal operations" can be performed with transfinite numbers. Analogous operations can be defined, but the are not the SAME operation. The fact that transfinite numbers have mathematical properties has no bearing on whether or not they have a referent in the real world - mathematics deals with lots of things that are pure abstraction with no actual referent (look into abstract algebra). — Relativist

I did not claim that because transfinite numbers have mathematical properties they can have real world referents. I don't see where in the part you quoted of me indicates that, the part you quoted was my response to a user claiming infinite numbers "act non-numerically". Yes, I've studied abstract algebra, I never claimed all mathematics was applied math.

That said, addition and multiplication can still be done with transfinite numbers. Cantor himself showed this, so it's old hat. If you mean that it's not literally the same operation, I'm just questioning the relevance. Transfinite arithmetic is arithmetic for infinite numbers. Is it a bit different? Yeah, but I never said otherwise. My point was that the results are odd because you're not in a finite domain anymore, e.g. ℵ0 + 8 = ℵ0.

It would be like looking at negation in paraconsistent logic and saying "Hey that's not negation because it isn't explosive".

Whatever man, I've said my piece in this thread as many times as I feel like doing anymore. Responses that it's magic or that it's contradictory or that it's nonsense aren't good responses when those declarations are only at best backed up by using definitions besides the actual definitions mathematicians use.

So... I'll leave you to it.