But maths has a responsibility to make sure it clearly communicates concepts to its end users. — Devans99

It does. Literally every mathematician and any maths student paying attention knows the things I'm saying.

Actual Infinity need to come with a health warning:

1) This is a conceptual concept only
2) Applying it to the real world is nonsense
3) It is logically inconsistent with the rest of maths and common sense (see Hilberts Hotel) — Devans99

#1 is not known to be true and previous examples were given of it possible being false.
#2 is the same as before.

#3 is just stupid. It's not logically inconsistent at all, stop saying things you've been rebutted on and refuse to give real evidence of. I don't care about common sense, we don't deal with infinity in common matters so why should I apply it to the mathematics of infinity? Hilbert's Hotel is not a contradiction. No one thinks the scenario described can actually occur. The process of moving hotel guests around is a temporal process that operates at finite speeds. We only have a finite amount of material, we can only create finite structures at finite rates over a period of time. Obviously it's never going to terminate for those reasons. None of those things are true of mathematical operations involving infinite sets.

It doesn't behave non-numerically, that doesn't make sense. The normal operations can be performed with such numbers, but that doesn't mean you'll get the results you would expect with finite numbers. And the reason is clear: Because you're dealing with a different type of number.

Whether such numbers have any place in science is an empirical matter, not an a priori one.

These properties are nonsensical compared to the properties of any normal number.

X+1=X

No other number, complex, vector, matrix, whatever, has this nonsensical property. — Devans99

I repeat, you can say that about ANY other kind of number that has a unique property others don't have. It's not a contradiction so the repeated claims you made that it was contradictory are just false. Weird is not the same as false, neither is unexpected, nor is counterintuitive.

Well transfinite cardinals have strange properties like:
X+1=X
X-1=X
What sort of number behaves like this? — Devans99

Obviously they have different properties. Finite numbers are finite, transfinite numbers are infinite. It's like complaining that odd numbers cannot exist because you can't divide them by 2 with no remainder like you can with even numbers.

And which numbers behave like that? I've already answered this: Transfinite cardinals and transfinite ordinals behave like that. It's only finite numbers which change in size by removing finite amounts from them.

This definition is self contradictory; the cardinality of the natural numbers is not a number as it claims. — Devans99

It's not contradictory. How many times are you going to make wild claims with no explanation?

How is it not a number? Cardinal numbers are numbers. Transfinite cardinals are cardinals. Therefore transfinite cardinals are numbers. Seriously man, go read some actual foundational mathematics stuff. You're wasting everyone's time by ignoring the actual learning needed to even understand the fundamental terminology at play. Linking Wikipedia articles and providing simplistic arguments which ignore the actual definitions actual mathematicians use for these words is lazy and deceptive. You're not doing philosophy, you're being an ideologue.

If it's an object then it exists. To be an object is to exist. There is no non-existent object, that's contradiction. You're just trying to find a semantic loophole, but you are really digging yourself deeper into a hole of contradiction. — Metaphysician Undercover

What one quantifies over in logic or places in a set does not commit one to the existence of the thing. If you think modern logic assumes existential import, then, well you just believe something false. It's not a "semantic loophole", you're just wrong man.

Our disagreement is over what constitutes a collection. I think that things must be collected to be a collection. You seem to think that things which are by some principle "collectible" are a collection. Clearly you are wrong, a collection must be collected, and collectible things do not constitute a collection. "Collection" often refers to the act of collecting. So it is quite clear that a collection does not exist until the things are collected in the act of collection. That is why an infinite set is utter nonsense, because it is absolutely impossible to collect an infinity of things. — Metaphysician Undercover

"Collection" does not refer the process of collecting things. If I talk about the collection of stars in the sky and I call that a set, no one thinks I've literally gathered the stars in the sky. They readily understand I'm mean that there's a condition each of those objects share (that is, "being in the sky") and that I'm grouping them into a collection.

And again, this is literally just an argument by definition, easily defeatable. Let's say I'm not talking about "collections" because you're somehow just obviously using the only coherent, sensible definition of that word. Here's a new word: "schmollections". Schmollections are like collections, except they don't refer to the process of collecting things. They refer to well defined groups of objects related by some common property, condition or rule and are referred to as a whole as a "Schmet" because OBVIOUSLY that's not a "set", supposedly. Well great then, it looks like infinite "schmollections" and "schmets" are possible since they don't make the same assumptions as "collections" and "sets" according to you. So modern mathematics (which you are dispensing with by making these objections, funnily enough) use "schmollections" and "schmets".

You're argument is trivial and presumptive.

Absolutely not. Until you demonstrate how an infinity of things may be collected into a collection, your definition of sets does not allow for an infinite set. You are in denial, refusing to understand the words of your own definition. — Metaphysician Undercover

You are confusing your own definitions with the definitions used in mathematics. I an not. I've stated my definitions, you're only response was to equivocate by making objections from your own separate definitions. I've already shown how easy intensional definitions of sets allows for perfectly obvious infinite sets to be created.

No, clearly we agree on the definition of a set, it is a collection. But nothing can belong to a collection without the act of collecting (collection), by which the thing is collected. And it's also very clear that an infinity of things cannot be collected. Therefore, according to the definition of 'set" which we both agree on, an infinite set is absolutely impossible — Metaphysician Undercover

No we clearly do not agree. You think collections are by definition finite, I do not. And unlike you, my definitions are actually used by virtually all modern mathematicians.

I can’t because cardinality is an undefined concept - it includes cardinality of actually infinite sets - which are not numbers so I can’t add one to it. — Devans99

It's not an undefined concept, you are out of your mind. Transfinite cardinal numbers are numbers which are infinite. Since wiki wish apparently good enough for you, here: http://en.wikipedia.org/wiki/Transfinite_number

Cardinality should be defined as the NUMBER of elements in a set. Actual infinity is not a NUMBER.

The cardinality of a set is determined by the number of elements of a set. The cardinality of the set {apple, banana, orange} is three because there are three elements. The cardinality of the set of natural numbers {0,1,2...} is aleph-null, the smallest infinite number. You have absolutely no idea what you're talking about.

Yes but my argument shows none of them exist because they are not constructable — Devans99

They are constructable. I gave an intensional definition of the sets. If you literally mean that "construct" means to individually gather each element and put them in a real box, well there's your problem. Your definition of set membership is inadequate for any useful mathematics beyond simple combinatoric reasoning.

- So you are saying there is a number (cardinality) with the property that number plus one equals same number? X+1=X !!! — Devans99

It's provably the case. Add one to the cardinality of the natural numbers. It's still able to be put into a function with the original set of natural numbers, and is still not uncountably infinite, as it cannot be put into a function with the set of real numbers. Arguments from incredulity don't impress anyone.

There are less squares than numbers because not all numbers are squares. Yet each number has a square so the number of numbers and squares must be the same.

He is trying to compare two actually infinite sets, IE comparing two undefined things. A set definition is not complete until all its members are interated. — Devans99

Again, a total misunderstanding. Stop linking things you don't understand. There's a one-to-one correspondence between the squares and the positive integers so we know those sets have the same cardinality. It only seems weird because you don't understand the properties of infinity. No contradictions arises here, and further infinite sets are not undefined. They can be given perfectly clear intensional definitions. Hell, you just gave an intensional definition of two infinite sets: the square numbers and the positive integers.

1) Actual Infinity is larger than any other number
2) Actual infinity plus one is larger than actual infinity
3) Hence there is no number larger than all other numbers
4) Actual Infinity does not exist — Devans99

#1 is false because there are lots of infinities, with some being larger and smaller relative to others. The infinity of the natural numbers has a smalelr cardinality than the infinity of the real numbers. There's no one singular "actual infinity".

#2 is wrong as well. Adding one member to an infinite set does not increase its cardinality (and neither does removing one element). This is bound up in the very definition of infinity, as it's finite things which change in size when things are added or removed.

#3 is just a complete misunderstanding of infinity in modern mathematics.

#4 is just what you've been assuming in every premise. The very misguided definition and misunderstandings of infinity are where the errors lay. Please just go study some set theory, stop this pseudo-philosophy of mathematics.

That's false. To put something into a set is to assign it some sort of existence. If Harry Potter characters are non-existent then the set of Harry Potter characters is an empty set. If you assert that the set of Harry Potter characters is not empty then you assert the existence of Harry Potter characters. — Metaphysician Undercover

That's just not true, no axiom in set theory entails this nor in classical first order logic. Objects quantified over are not assumed to exist. The set of Harry Potter characters has members, but the members do not exist as real things. To call it an empty is to say that there are Harry Potter characters, in which case the books are entirely gibberish.

Yes, as I explained, the definition of "set" precludes the possibility of an infinite set. A set is a collection. It is impossible to collect an infinite number of things Therefore an infinite set is impossible. Some people, like you, just like to deny the obvious. That means that you are in denial. — Metaphysician Undercover

The "collection" is not created through a temporal process. A collection does not entail finitude. Again, you are literally just proving my point: It's the definition you and only you are using, it is not the definition used in modern mathematics. A set is a well-defined collection, often characterized by sharing some property in common or holding to some specified rule. The set future moments is a perfectly comprehensible set, as is the set of natural numbers. Our condition for what makes certain things members of those sets is in the conditions themselves, (e.g. being after some specified time "t", or being a natural number (zero and greater)).

Well, so much for your "clear" definition of infinity in mathematics then. You seemed to be so certain of that point. I'm glad you now see that you were wrong about it. — Metaphysician Undercover

"My" definition (in actuality, the mathematical definition) of sets are clear and they allow for infinity. My point was there's no one definition to which you (as you did) can appeal to to claim that sets are "by definition" finite collections. You're just not wanting to acknowledge that such is a limitation of one definition of a set which you use, not the one mathematicians use.

As I said, that something has the property of being red, is a judgement. Whether an object is red or not requires a definition of "red", and a judgement as to whether the thing fulfills the criteria of being red. That definition, and that judgement, are necessary because "having the property of being red" is a relation between the universal "red", and the particular object which is said to be red. Otherwise "red" might be defined in any way, and any object might be red. Or do you think that "red" has determinate meaning without a definition? — Metaphysician Undercover

You are confusing determining if an object belongs to a set with whether or not the object does in fact belong to a set. Judgements are made by agents, sharing a property in common has no dependence on people's judgements. What the property "being red" corresponds to has nothing to do with people. Intensionaly defining sets is not about going about and determining what specific objects go in the set, it's simply a way of specifying what criteria is entails being a member of some set. Like if I say "the set of even numbers" the objects which satisfy this can be determined, but it's not like I actually have to ascend up the natural numbers to know which ones will be in the set. It's all in the definition, I already know what makes a number an even number.

It seems like you're redefining "set' to suit your purpose. No longer does "set" refer to a collection, it refers to things which are collectible, potentially collected. That's the issue of the thread, things which can be potentially collected together do not make an actual collection. And in the case of infinity, an infinite number cannot even be potentially collected together, because the definition of infinity makes collecting an infinite number impossible. So all you are doing with your "infinite set" is asserting that the impossible is possible. That's nonsense. — Metaphysician Undercover

You are making up definitions of sets, I'm literally using the standard mathematical definition which in fact captures many of our intuitions about collections and does so without any contradictions. It's not about being potentially collected (whatever that means, sounds like you're again assuming everyone is using your definition). What I am asserting is that sets are well-defined collections which can be defined intensionally or extensionally. The former allows one to easily define infinite sets without any contradictions. It's not asserting the impossible, it literally has (as I've already given) perfectly clear examples which are infinite and which to not result in any contradictions within the standard math formalism. Show the contradiction from the actual mathematical definition of a set or else you're just ignoring mathematics.

Discribing the set is not the same as the set itself. The description is incomplete unless all members are iterated. — Devans99

Describing a set is how you populate it with members, e.g. Set "A"={1,2,3}

The definition of a set from Wikipedia starts with:

‘In mathematics, a set is a collection of distinct objects, considered as an object in its own right.’ — Devans99

Why don't we look a bit further than the first sentence, yeah?

A set is a well-defined collection of distinct objects. The objects that make up a set

[...]

There are two ways of describing, or specifying the members of, a set. One way is by intensional definition, using a rule or semantic description:

A is the set whose members are the first four positive integers.
B is the set of colors of the French flag.

The second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets

Sets are not like baskets, I don't need to engage in a temporal process in order to "make" a set. If I talk about the set of red things, it's not like I had to go out and get all the red things and put them somewhere, my specification covers them all immediately. And so too with infinite sets.

The set of natural numbers is not constructable through any known operation hence it does not exist as a completed set. — Devans99

Sure, if we just ignore standard mathematics you can believe that.

You have given no evidence of that and you have fundamentally misunderstood many aspects of these issues. Transfinite cardinals and ordinals are infinite numbers so an actual infinity is perfectly coherent. Not a single known contradiction can be derived in standardly studied mathematical formalisms containing the Axiom of infinity. And infinite sets are defined, not undefined.

This has been a waste of time IMO. You haven't dealt with the actual definitions and elucidations of these things as done in modern mathematics.

You can’t take one from from Undefined — Devans99

It's not undefined, it is literally defined.

But my point is actual infinity is not a number so Actual Infinity is undefined in mathematics (except in set theory which merely declares that it exists as an axiom). — Devans99

Again, false. Infinity is not a natural number, but there are many kinds of infinite numbers. Namely, those which are the cardinalities of the innumerable infinite sets. The natural numbers have a set size of aleph-null. Take one member out of that set and it's size is still the infinite number aleph-null. Transfinite cardinals and ordinals are infinite numbers, so you're just wrong.

There is no natural number with the property that yoacan keep subtracting one from and never reach zero. Hence actual infinity does not exist. — Devans99

That's just a misunderstanding. Infinity is not a member of the set of natural numbers, so of course there's no natural number of which you can indefinitely subtract from without reaching zero. But the cardinality of the set of natural numbers will.never reach zero just by subtracting members from the set. So your conclusion does not follow.

Calculus resolves Zeno’s paradoxes in a complex way.
- Denying Absolute Infinity (and thus implying discrete time) solves them in a simple way
- it also solves the other paradoxes of infinity (https://en.m.wikipedia.org/wiki/Paradoxes_of_infinity)
- Occams Razor simple solutions are better than complex ones. — Devans99

They are not on par. Occam's razor is to be used when all else is equal. Denying infinite time is not simple, there's really no independent reason to posit time as finitely indivisible, so it's just unnecessary since we can eliminate most so-called paradoxes involving infinity.

What annoys me is that bringing Zeno up so much makes it seem like there's no standard resolution to most of Zeno's paradoxes, which is just ignoring the calculus.

I've no clue if she created it, but my grandmother used to say:

If you do something stupid, don't do it twice.

I suppose I was never good at putting that into practice though...

I’m not saying maths is incoherent, just pointing out it’s impossible to define the cardinality of an infinite set so maybe infinite set is a flawed concept as was argued earlier... — Devans99

That is saying standard mathematics is incoherent. Standard mathematics incorporates multiple levels of infinity with different cardinalities. It's not impossible to define said cardinalities, I did so in previous posts. A wholesale denial of the coherence of defining the Cardinality of infinite sets represents and abandoning of standard mathematical formalisms, even the non-classical ones.

Actual Infinity
=
Cardinality of the set of natural numbers
=
Nonsense — Devans99

So you are going even further than limiting infinity in physics and just denying the coherence of standard mathematics.

There is no such "set". The moments after the present moment have not yet come into existence so you cannot collect them into a set, nor can they be members of "a set" in any way or fashion, as they are non-existent. You are claiming to have a set of things which do not exist, but that's impossible so it's pure fiction, nonsense. — Metaphysician Undercover

Putting non-existent things in a set in no way commits one to their existence (goodbye existential import). The set of Harry Potter characters is only populated by non-existent things.

It's a matter of understanding the definition of "set", and understanding the definition of "infinite", and realizing that it is impossible to have an infinite set. These two are incompatible, by definition, so talking about infinite sets is contradictory nonsense. Of course we all know that because of the many paradoxes which are known to arise from the assumption of infinite sets, but some like you, choose to ignore this obvious fact. — Metaphysician Undercover

It's question begging because no one is using your definition of infinity which is defined in a way so as to preclude being actual, nor does the definition of a set preclude it from being infinite. There's no understanding "the" definition because there is no one definition. The maths definition of infinity is actually useful since it's crucial to modern mathematics (see calculus) and introduces no contradictions. There are no paradoxes involving the mere concept of infinity in standard mathematics, otherwise you could provide the proof of such a contradiction from the axioms and inference rules in the standard formalism.

That something has a particular property is a judgement. The thing is a particular the property is a universal. Therefore if "sharing a property" is what is required to be a member of a set, then a judgement is required in order that things be of the same set. So the declaration "you go in this set" is exactly what is required in order that a thing be a member of a particular set. — Metaphysician Undercover

Incorrect. If two things hare a property they share it whether or not I judge them to. Two red objects share the property of being red even if no one exists to recognize such. So to speak of sets having members based on a shared property in no way requires a judgement to make it so.

You seem to either believe that sets just naturally exist without ever being created by human minds, or else that things automatically jump up and join any set which they are supposed to be a member of, without being counted into that set. So either the green grass is naturally a member of the set of green things without that set ever being created by a human mind, or else the green grass jumps into the set, of its own power, as soon as "the set of green things" is named by a human being. Both of these, I tell you are nonsense. — Metaphysician Undercover

You're doing it again. It's not a mechanistic process that occurs over time nor is it necessarily done by an agent. Sets don't exist in the mind. The "set of numbers greater than 500 trillion but smaller than 1 quadrillion" is simply too large to be conceptualized in the mind, but it's obviously a perfectly legitimate set.

As a space, the continuity is about the ability to maintain certain general symmetries rather than any physical continuity as such. — apokrisis

I didn't make any point regarding physical continuity (if space can even be called physical).

To say the Universe is just "actually infinite" is hollow metaphysics - a way to avoid the interesting questions. What came before the Big Bang? Where does the Cosmos end? You seem to want to shrug your shoulders and say everything extends forever. That is what maths would say. So let's just pretend that is the case. — apokrisis

I don't think the universe is actually infinite in breadth or in the past, I really have no idea. And I'm certainly not saying such questions should be shrugged at. From the very beginning is took issue with the OP's assumption that any sort of actual infinity was impossible in virtue of pure logic (because, supposedly, contradictions crop up). The only points I made about maths were in support of that point. We know no known contradictions are derivable from employing infinity in standard maths. So it doesn't make sense to say actual infinities are impossible because of an inconsistency. They may well be impossible, but as I started by saying that isn't because of anything regarding inconsistency:

You might say that not every aspect of our particular universe can be infinitized, but there's no argument that the concept itself precludes instantiation in the world. — MindForged

The infinite set is specifically designed for no reason other than to break this law, therefore it is unreasonable, nonsense. — Metaphysician Undercover

It's not "designed" to break this "law", it just doesn't apply and it's perfectly obvious that it wouldn't. These sets aren't conjured, they're the numbers we start of learning.

As it is an unbounded (open) set, it is not truly a "set", as a collection of objects, it is a boundless collection which is not a collection at all. — Metaphysician Undercover

Again, what is the non-question begging argument for this? What makes a set a set is not it being bounded. The "set of moments after the present moment" is unbounded but no one gets up in arms about defining such a collection of moments as a set. They share a property in common (their coming after the stipulated moment) so assigning them to the same collection is natural.

A collection, or "set" means that the members are collected together in a group. If the collecting is not complete, then the described collection (set) does not exist. To call it a collection, or set, is contradictory nonsense. — Metaphysician Undercover

They aren't "collected" in a mechanistic process, i.e. going out and declaring "You go in this set" and such. Just sharing a property is enough, and it happens to be perfectly compatible with there being infinite collections.

Hell, let's just show this without reference to numbers as elements of a set and yet it still be infinite:

Let P be the set of all possible English sentences.

It's surely unbounded. I can always add a new word to any English sentence to yield evermore new sentences and it's still going to fall in the set of "possible English sentences". And yet there's no way you can argue it fails to be a set in virtue of being unbounded.

You have demonstrated that the set of natural numbers is equivalent (in the sense of having the same number of members) as the set of even numbers. That's nonsense, and that's what the concept of infinity introduces into mathematics, nonsense. — Metaphysician Undercover

Nonsense based on what argument? This is what you say but:

It's nonsense because it's a totally useless piece of trivia. Infinite sets have the same number of members as other infinite sets ... a nonsense number ... an infinite number. — Metaphysician Undercover

Like how is the a an actual objection? It's "nonsense because it's useless trivia". Come on, it's literally a property by which we can clearly distinguish one type of set (finite sets) from another type of set (infinite sets). All you're doing is saying infinity is nonsense but you're not actually explaining why.

And you have been reminded a few times that these solid theories in fact depend on working around the infinities they might otherwise produce. So it ain't as simple as you are suggesting. — apokrisis

That's not true, using an infinity is not the same as a singularity occurring in the theory. Space under relativity is treated as a continuum, but that's not the same as a singularity occurring, it's just part of the geometry.

You seem to think I'm arguing that anytime infinity crops up in our models it ought to be accepted. As I said initially, we have good reasons why we don't do that (the need to get meaningful results being central). But my point was that we still make assumptions (crucial, necessary ones) regarding the existence of infinity in the world as well (relativity and QM both do so), so the notion of an Actual Infinity isn't off the table.

Instead of being fundamental, the perfect regularity and simplicity of a classical geometry is the most exceptional case. It requires a lot of explanation in terms of what removes all the possible curvature, divergence, and other non-linearities. — apokrisis

I certainly haven't said Euclidean geometry is how our universe is actually structured. I said the opposite, in fact.

I see no clear definition of infinity here, just a rambling description of a particular type of set, which you call an infinite set. That description doesn't tell me what it means to be infinite, it tells me what it means to be an infinite set. — Metaphysician Undercover

It is a "particular type of set" which distinguishes the finite sets from the infinite ones by means of a relationship that isn't possible for finite sets. It further allows us to see the exact difference between such sets. A subset is a "proper subset" of a set so long as the members each contain are not all identical, but some are shared. For finite sets, proper subset will always be non-identical and leave some out of the original set. But for an infinite set, this cannot happen, just look (Naturals on the left, evens on the right):

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

There's never a point at which the one-to-one correspondence fails to pair up a natural with an even. We know the evens are are proper part of the naturals, as the evens are lacking half the naturals (the odds). And yet they have the same cardinality. That's infinite and it returns exactly the sets of numbers we already intuitively take to be infinite, and (as I said) it gives us a property by which to tell which is which and does not yield any contradictions.

How this is rambling, I don't know. It's literally just lining things up.

We can’t conceive of logically inconsistent concepts like Actual Infinity in a logically consistent way.

I’d allow for the existence of the inconceivable only if it where possible. No need to allow for impossibilities like Actual Infinity. — Devans99

Prove it. I've given evidence that we can conceive of the actual infinite by giving a description of it and examples which instantiate it, you just keep begging the question or just asserting what you believe. And of course we have need of the actual infinite. As has been said a few times, several very solid theories make assumptions that include infinity. And as to my original point, if you accept almost any fleshed out mathematics you have to accept that infinity is not a contradictory concept. So to say it's contradictory when applied to reality either makes no sense or you have an unstated argument.

I'm pretty sure I did in a previous post, but to take sets

A set is infinite if it's members can put into a one-to-one correspondence with a proper subset of itself. So we know the natural numbers are infinite because, for example, there's a function from a set to a proper subset (read: non-identical) of itself like the even numbers. For every natural number, you're always able to pair it up with an even number and there's no point at which one of the subset cannot be supplied to pair off with the members of the set of naturals.

That's pretty clear, it's exactly the same reason I can, without knowing the exact number of people in an audience, know that if every seat is occupied, then there's no empty seats (each seat can be paired off with a person).

that gives us a good reason for thinking of space and time as a continuum and no good reason for thinking otherwise. — SophistiCat

This is more or less what I use as justification. I wouldn't put it forward as unchallengable or something, but insofar as we accept what our best theories say I tend to informally just say they're true. I do however believe there are also arguments for the continuous nature of space that bolster that belief as well.

But it’s impossible for to construct a smallest possible distance (1/infinity) - we can merely construct successfully smaller distances in a process that tends to but never reaches 1/infinity. That’s the definition of potentially infinite. I asked for an example from nature that is actually infinite... — Devans99

There's no "constructing" here, space is just infinitely divisible. There's no such thing as a smallest possible distance.

If it is, it’s a potential infinity rather than an actual Infinity (you do understand the distinction?).

The division of space takes time, first we must cut one inch, then 1/2 an inch, then 1/4... No matter how many cuts we make we never get to actual infinity, just some small number. — Devans99

I understand the distinction, you do not understand the point. I'm not talking about the temporal process of looking at ever smaller slices of space. I'm saying that the nature of space itself is such that it is infinitely divisible already; for any two points in space there are in actuality points in between them. You'll never reach a base unit of space because no such thing exists, it's a continuum.

I'm not a Pythagorean, that's just a silly response. Even if I were a mathematical platonist, nothing about that in particular makes me think some aspects of reality might be infinite. Besides, I've given you one example already: Space is infinitely divisible. This is born out in Relativity and in relativistic quantum mechanics (probably in non-relativistic QM too, but I've only doen classical mechanics) , and there are even good strictly logical arguments for the infinite divisibility of space. That aside, my constant objection has been that you haven't given a single argument against the possibility of actual infinities that was question begging. Look:

I believe that leads to contradictions. For example, how could we ever reach today if the past stretches to negative infinity — Devans99

Not a contradiction.

Time however is part of the physical universe so it can’t be actually infinite. — Devans99

Assuming the thing being discussed, namely, that something in the world could be infinite.

That's all you've really done so far, these aren't serious arguments IMO.

Numbers aren't part of the mind. And besides which, you're contradicting yourself. Previously you said that the mind was finite. But you're saying numbers can be infinite, yet you also said numbers are figments of our minds (which are finite). That's just inconsistent. Anyway, you haven't given an argument for why time can be infinite, you're just presupposing it.

There's no force to this objection. We're not at an end point. Ignoring our actual universe, if the past were infinite then we would just be at an arbitrary part of the sequence of time. I don't see the issue of "reaching today" any more than it's a problem that I can reach 100 despite there being an infinite number of real numbers (decimal numbers) between 0 and 1 alone.

The above refers to future which is potentially infinite which is not the subject of this thread. — Devans99

Did you miss the word "before"? That was talking about a past series of infinite moments.

Past infinite time is however an Actual Infinity so is disallowed. For example this argument:

- Time is a series of moments
- The moments so far must be an actual number not infinity
- So time has a start — Devans99

Premise 2 is the obviously question begging premise. Nothing about the concept of "moments of time" precludes an infinite past series of moments. I repeat:

For every moment before this very moment, there is another moment.

That might well be false in the universe we are in (it looks like it has a first moment of time), but the sentence entails no contradictions unless infinity is a contradictory concept; it isn't, ergo there are no inconsistencies.

It's an axiom because no one has found any contradiction that is provable from it. The definition of infinity is pretty clear, it's extremely useful in mathematics and science, and it introduces no contradictions into the theorems. If that makes a "shaky ground" I have no idea what use your standards of a good foundation is to anyone.

I don't see how that's a given. Space is infinitely divisible. Whether or not space counts as a "thing" or not I don't think matters, but it's infinite.