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.

Yes, only somebody that does not understand mathematics or logic could believe there are contradictions. It is a matter of personal taste whether one accepts it, a bit like the axiom of choice. But as you say, the cost of rejecting the Axiom of Infinity is very high, much higher than rejecting the axiom of choice.

That is not the mathematical definition of a set. The mathematical definition of a set is that it obeys the axioms of the set theory in which we are working. The most commonly-used set of axioms is Zermelo-Frankel - ZF. The concept of 'collection' does not form part of those axioms.

But even if we were to try to use the definition you suggest, it would be incorrect to say that infinite sets are not well-defined. In mathematics the words 'well-defined' have a very specific meaning, and they only apply to functions, not properties (aka relations). We say that a function is 'well-defined' if, using the definition to apply it to an element of its domain, there is a unique object that is the image of that element under that function. The notion of being a set, or of having finite cardinality, is a property, not a function, so the notion of 'well-defined' is not relevant. — andrewk

Thanks for the information. I know that mathematical definitions are called axioms. The problem that I have been trying to shed light on, is that some axioms contradict other axioms, so that within the field of mathematics in general, there are contradictory axioms. To me, this indicates that the principles upon which these axioms are founded, are not well understood. This is most noticeable in concepts like "infinity", and "zero", but in some cases in modern physics I've noticed that it extends into geometry and dimensionality as well. One case discussed here already at tpf is Euclid's parallel postulate.

If you really dislike the concept of infinity, all you need do is reject the 'Axiom of Infinity', which asserts the existence of a set that can be thought of as the set of natural numbers. Without such an axiom, we can have natural numbers as large as we wish, but there is no such thing as the set of all natural numbers. Such an approach to mathematics is consistent, and some people try to limit themselves to that. The trouble is that it is that axiom that gives us the tool of Proof by Mathematical Induction. Without it, there is an enormous volume of important results that we would not be able to us. — andrewk

Yes, the easy way out would be to reject the axiom of infinity. But to reject an accepted convention without good reason, is generally considered as being irrational. That's why we ought not simply reject the axiom unless it has been demonstrated to be unacceptable.

I do not agree with you about the relation between the axiom of infinity and mathematical induction though. All that mathematical induction requires is that what is true of one number is true of the next, and therefore true of all the following numbers. It doesn't require that the numbers are infinite, because it works also in a descending order, in which there is an end, a lowest number. That the natural numbers are infinite is a separate axiom, unrelated to mathematical induction. We can have an axiom which states that the natural numbers are infinite, without allowing that the natural numbers are a set.

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. — MindForged

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.

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".

Hi there, l refute the two arguments in the OP:

* Can't reach infinity, so actual infinity is impossible
Answer: impossible for you to attain to yes, therefore actual infinity is preeminent & the primordial state. That's also a powerful argument for the Big Bang Universe being finite. One doesn't just grow into infinity. [ofc a finite universe is impossible, because it'd have borders with nothingness, which is the logical definition of crazy, absurd - all that really exists is the primordial state, which is actual infinity i.e. BBT is true & proves universe is imaginary]

* Zeno's Paradox: No problem, Achilles takes however many steps to reach the tortoise. "Infinitely small" is defined as being within those steps, i.e. it's predicated on those steps being made. SO ... let the steps be made. No problemo!


[It's not that actual infinity is being achieved by Achilles. The divisions are conceptual, or at the very least, they aren't one distance, therefore they aren't an infinite distance, therefore Achilles isn't crossing an infinite distance. It's infinite conceptual divisions, moreover, each division keeps reducing the distance. So, linguistically (?) speaking: an (an = one) infinite distance (it's not a measure of distance) is NOT being crossed by Achilles. Now, have Achilles stop to catalogue the infinite subdivisions - forget Planck - that would be attaining to actual infinity. I believe in an actual infinity btw!]

All that mathematical induction requires is that what is true of one number is true of the next, and therefore true of all the following numbers — MU

This works for a logical theory in which the only objects in the domain of discourse are natural numbers. In that case, we can just use the following axiom of induction:
$(P(a)\wedge \forall x(P(x)\to P(x+1)))\to ((x\ge a)\to P(x))$
and that does not require any assertions about infinity.

However this doesn't work if we want to have objects in our domain of discourse other than natural numbers, because then we need to add a condition 'x is a natural number' to the above induction axiom, which requires referring to the set of natural numbers, whose existence cannot be asserted without the axiom of infinity, or some equivalent..

I think this issue of the axiom of infinity may be related to that of omega-completeness, which is about whether there may be natural numbers other than those we get by adding 1 to 0 a finite number of times, ie 'non-standard' natural numbers. Omega-completeness is a very interesting subject, but it usually gets my head all muddled when I try to think about it, if I haven't done so in recent times.

— tim wood

There cannot have not been infinitely many paths TAKEN, there are only infinitely many possible paths that could potentially be taken, but it is impossible to actually follow them - no matter how long we have to try. So these paths exist in the abstract, but not in the real world. — Relativist

Why do they have to be taken to be real? If they're not taken are they not real? .... Each is a possible path. It's the "taken" you object to? But whenever was a clock attached to a number?

Arguably: no, they aren't real.

For starters, look at this physically: it is impossible to determine a position with any precision smaller than a Plank length (1.6 x 10^-35 meters.) Therefore there is a minimum width for possible paths, and thus the number of paths that could possibly be taken is finite. (source).

But what I'm actually objecting to the treatment of abstractions as existents. Triangles do not exist; rather: objects with triangular shapes exist. We form abstractions by contemplating objects with similar features and mentally omitting the non-common features. Philosophers call this the "way of abstraction"). This is not actual existence. For this reason, it is inadequate to point to abstractions (or mental objects) as examples of actual infinities. If someone can come up with an example of an infinity in the real world, that is not just a mental object, then I'd jump on the pro-infinity bandwagon.

But even if we were to try to use the definition you suggest, it would be incorrect to say that infinite sets are not well-defined. In mathematics the words 'well-defined' have a very specific meaning, and they only apply to functions, not properties (aka relations) — andrewk

Ok let’s use the language ‘fully defined’. A set is only fully defined once we have listed all its members. Clearly infinite sets are not fully defined yet maths tries to treat them in the same way as a finite set (which is fully defined).

The trouble is that it is that axiom that gives us the tool of Proof by Mathematical Induction. — andrewk

We just need an axiom to the effect that ‘the natural numbers exist but not as a completed set’ and Induction still holds.

finite universe is impossible, because it'd have borders with nothingness, which is the logical definition of crazy — SnoringKitten

That’s not crazy; just think about spacetime; where it is not, time does not exist so there is absolute nothingness (in contrast to everyday empty space which has vacuum energy and a time coordinate).

Now, have Achilles stop to catalogue the infinite subdivisions - forget Planck - that would be attaining to actual infinity. I believe in an actual infinity btw! — SnoringKitten

The ‘infinite’ points on a line segment is not an example of the actually infinite. If you use a sensible definition of a point (length >0), then there are always a finite number of points on a line segment and it is an example of potential not actual infinity.

Ok let’s use the language ‘fully defined’. A set is only fully defined once we have listed all its members. Clearly infinite sets are not fully defined yet maths tries to treat them in the same way as a finite set (which is fully defined). — Devans99

You have defined a new term in relation to sets - 'fully defined'. What then?

It’s the set concept that is flawed. Maths uses a polymorphic definition of the word ‘set’:

- a collection of distinct objects like {1,2,3}
OR
- the selection criteria to populate a set like ‘the natural numbers’

So no wonder set theory is confusing with such an anomaly at its core.

The two different types of ‘set’ have different properties. One does not have a cardinality or a complete list of elements for example.

Making up magic numbers for the missing property (cardinality) is not the correct approach. Rather set theory should recognise these sets are two very different objects with different properties.

So a fully defined set has a cardinaity
But the selection criteria for a set does not.

How about this approach:

- An Actual Set is a collection of distinct, listed objects like {1,2,3}
- A Potential Set is the description of a potentially collectable set of objects.

So maths can still talk about the ‘potential’ set of real numbers...

I think the mathematicians have the definition of Point wrong:

“That is, a point is defined only by some properties, called axioms, that it must satisfy. In particular, the geometric points do not have any length, area, volume or any other dimensional attribute’

https://en.m.wikipedia.org/wiki/Point_(geometry)

If a point has no length it does not exist so the definition is contradictory.

A point must has length > 0 else it does not exist. With this revised definition of a point we can see that the number of points on any line segment is always a finite number rather than Actual Infinity. — Devans99

See the section "Mathematical Points and Dimensional Points" at:
http://www.theintermind.com/ExploringThe4thDimensionUsingAnimations/ExploringThe4thDimensionUsingAnimations.asp You will have to scroll down a bit to get to it.

However this doesn't work if we want to have objects in our domain of discourse other than natural numbers, because then we need to add a condition 'x is a natural number' to the above induction axiom, which requires referring to the set of natural numbers, whose existence cannot be asserted without the axiom of infinity, or some equivalent.. — andrewk

I think that's all well and good, because no two objects are exactly the same, and the difference between this object and that object is never the same as the difference between that object and another object, contrary to what is the case with numbers. This provides us with a separation between the "ideal realm" of numbers, in which perfection is the ruling principle, and the physical realm of objects, in which the uniqueness of the particular is the ruling principle. Therefore we could have a principle whereby what is true of numbers is not necessarily true of objects.

Consider that the axiom which dictates that the natural numbers are infinite, is very good and useful, because it allows us to count anything and everything. It is very well designed (if "designed" is the right word, because I don't know how it came about, it's just kind of intuitive) because no matter how many sets of things, or individual things we encounter, the numbers can always go higher than the number of things, allowing us to count more things as we encounter them, because numbers are infinite and things are finite. However, if we allow that the numbers themselves are things, as set theory implies, then we have encountered a type of thing which cannot be counted. If we try to count the numbers, there's always more, and we're thwarted. The consequence of this axiom of set theory therefore, which implies that the numbers are objects, is to negate the good and usefulness of the intuitive axiom which stipulates that the natural numbers are infinite. When it is allowed, as a principle in your axiomatic system, that there are objects which are infinite, you lose the capacity to measure all objects, which is what the axiom that the natural numbers are infinite gives us.

I think this issue of the axiom of infinity may be related to that of omega-completeness, which is about whether there may be natural numbers other than those we get by adding 1 to 0 a finite number of times, ie 'non-standard' natural numbers. Omega-completeness is a very interesting subject, but it usually gets my head all muddled when I try to think about it, if I haven't done so in recent times. — andrewk

It appears like the question of Omega-completeness is just an issue of whether the natural numbers ought to be consider infinite or not. As I said above, I think there is a very good reason to allow infinity for the natural numbers. Where I find a problem is in the idea that the natural numbers are objects, which is the idea that set theory builds on. This causes us to get muddled by the idea of infinite objects, and these are inherently unintelligible. One solution may be that of Omega-completeness, denying that the natural numbers are infinite/ But I don't think that's the best solution.

Imagine a Square drawn on a piece of paper. Now imagine the Square shrinking smaller and smaller. It remains a Square no matter how small it shrinks. If we stop shrinking it and start magnifying it back we can bring the Square back to the original size. But now imagine the Square shrinking to Zero size. All points of the Square collapse to a single point and there is no longer a Square on the paper. The square has been transformed into a single point. We would not be able to magnify the resulting point back the the original Square. We could also shrink a Triangle in the same way and at Zero size it would be a single point just like the Square. The Square and the Triangle lose their identity when they are Zero size. They become something different. They become something less than what they were. Zero size is an unrecoverable threshold of size that changes everything.

Now imagine a Square that is the smallest Square that is not equal to Zero. This thought sends your mind into an endless recursive loop of the Square getting smaller and smaller and we soon realize that it is impossible to imagine such a smallest Square. One thing we can say is that this Square is Infinitely small but is still a Square. In general mathematics this would be called a differential Square or an infinitesimal Square.

Next imagine the Square that was drawn on the paper growing larger and larger. If the Square was exactly in the center of the paper the sides of the Square would eventually move off of the paper and past the edges of the universe. It remains a Square no matter how large it grows. If we stop growing it and start shrinking it back we can bring the Square back to the original size. But now imagine the Square growing to Infinite size. The sides would all move out to infinity. No matter how far you went in the universe you would never encounter a side of the Square. The Square has effectively exited the universe. We could also grow a Triangle in the same way and at Infinite size it will no longer be found in the universe. The Square and the Triangle lose their identity when they are Infinite size. They become something different. Paradoxically they become something less than what they were. You might think that the Square and Triangle are still out there at Infinity. But there is no "there" at Infinity. The Square and Triangle are gone. If you think you can go out "there" to an edge of the Square or Triangle at Infinity then that "there" is not Infinity. Infinite size is an unrecoverable threshold of size that changes everything.

Now imagine a Square that is the largest Square that is not equal to Infinity. Similar to the differential Square, this thought sends your mind into an endless recursive loop of the Square getting larger and larger and we again soon realize that it is impossible to imagine such a largest Square. We can say that this Square is Infinitely large but is still a Square that exists in the universe.

I think that just as Infinite Squares are not possible it is probably true that any Infinite Physical quantity of anything is not possible. Just because an equation in Science goes to Infinity, it doesn't mean that the Physical quantity in the equation is able go to Infinity. I think this is a limitation of what we can do with Mathematics. Seems like a minor limitation but it has big consequences when equations in Science go to Infinity.

Hi i'm sorry but your reply to my reply either contains no counter reason (e.g. where you say a finite universe is NOT crazy, it just isn't crazy, without explaining why. It IS crazy because there can be no border between something and nothing, as l've explained. The intrusion of nothing - the impossible - into something, is the definition of the untenable, the absurd, the crazy)

or (when you talk about finite points, and so on) literally, grammatically, makes no sense. Please re-state.

A finite universe is more likely than an infinite universe:

- We have empirical evidence for the finite
- We have no empirical evidence for the infinite

Give me an example of the actually infinite in nature.

Just a thought: The premise that we start with is that Before The Beginning there was Absolute Nothing. There was no Energy, and there was no Matter, and there was no Space. Matter is made out of Energy but it is still useful to think of them as separate things. The no Space part of this is the most difficult to grasp. We naturally think of Space as an ever present background reality that infinitely extends out in three dimensions. It boggles the mind to try to think of no Space or even a finite Space. But we can conceive of other Spaces than our 3 dimensional Space. So our Space could have been a 4 dimensional Space for example. A 4 dimensional Space is a very different Thing than a 3 dimensional Space. The point is that if Space can be different Things then Space is a Thing. And like all Things we conclude that Space can exist or not exist. It is not the always present background reality that we assume it is.

The premise that we start with is that Before The Beginning there was Absolute Nothing. There was no Energy, and there was no Matter, and there was no Space. — SteveKlinko

I don’t think you can start with that premise:

- Something can’t come from nothing
- So something must have always existed
- So the state of ‘Nothingness’ is impossible
- If something is permanent it must be timeless (proof: assume base reality existed eternally - the total number of particle collisions would be infinite - reductio ad absurdum)
- So base reality must be timeless (to avoid the infinities)
- Time was was created inside this base reality

The premise that we start with is that Before The Beginning there was Absolute Nothing. There was no Energy, and there was no Matter, and there was no Space. — SteveKlinko
I don’t think you can start with that premise:

- Something can’t come from nothing
- So something must have always existed
- So the state of ‘Nothingness’ is impossible
- If something is permanent it must be timeless (proof: assume base reality existed eternally - the total number of particle collisions would be infinite - reductio ad absurdum)
- So base reality must be timeless (to avoid the infinities)
- Time was was created inside this base reality — Devans99

"Something can't come from nothing" is an unproven Belief when it comes to the beginning of everything. No one knows what happened back then. Science is pretty sure that the normal rules of Physics don't even apply. The things you say above probably don't apply. The rules of Physics don't apply because Science does not know all the rules of Physics yet. In any case, the concept of Nothingness is a real possibility. Why is there Something? The real point of the Thought experiment was to talk about Multi-Dimensional Spaces. The usual common sense Belief is that Space is some Background ever present thing that is always there extending out Infinitely in 3 directions. This is just because we live in a 3D Space and it is all we know. Space could have been 4D which is a whole different Thing than 3D. Where would all that extra Space in 4D come from?

Something can't come from nothing" is an unproven Belief when it comes to the beginning of everything. — SteveKlinko

I’m basing my argument on common sense and naturalism - not referencing any particular rule of physics.

- if you define nothing as no matter, energy, space or dimensions
- then it’s pretty clear ‘can’t get something from nothing’ holds
- so it follows something has existed always

Something can't come from nothing" is an unproven Belief when it comes to the beginning of everything. — SteveKlinko
I’m basing my argument on common sense and naturalism - not referencing any particular rule of physics.

- if you define nothing as no matter, energy, space or dimensions
- then it’s pretty clear ‘can’t get something from nothing’ holds
- so it follows something has existed always — Devans99

We don't even really know what any kind of Something is. You could be right. But when it comes to before the beginning nobody knows anything. What is this Naturalism? How do you know Naturalism holds before the beginning. We are completely Ignorant of beginnings. We can only Speculate and one Speculation is as good as any other.

What is this Naturalism? How do you know Naturalism holds before the beginning. — SteveKlinko

Naturalism is the exclusion of magic from our consideration of the physical sciences.

I assert that ‘something from nothing’ is a magical proposition so we can exclude from our investigations of the origin of things.

- if you define nothing as no matter, energy, space or dimensions
- then it’s pretty clear ‘can’t get something from nothing’ holds
- so it follows something has existed always — Devans99

If you define.... And if I define it as turtles all the way down, then it must be turtles all the way down, yes? What has the nothing of world - the universe - got to do with any definition or understanding of yours of nothing?

Your "common sense" and "naturalism" are the toys of children.

I assert that ‘something from nothing’ is a magical proposition so we can exclude from our investigations of the origin of things. — Devans99

All right, something from something, or no from, just the something. Care to tell us how old that something is?

But when it comes to before the beginning nobody knows anything. — SteveKlinko

That's self-contradictory. A beginning has no predecessor, or it's not the beginning.

Care to tell us how old that something is? — tim wood

I’m not sure it makes sense to talk about how old the something (base reality) is in the context of this argument. Remember the rest of the argument says base reality is timeless and permanent and contains time.

Now imagine a Square that is the smallest Square that is not equal to Zero. This thought sends your mind into an endless recursive loop of the Square getting smaller and smaller and we soon realize that it is impossible to imagine such a smallest Square. One thing we can say is that this Square is Infinitely small but is still a Square. In general mathematics this would be called a differential Square or an infinitesimal Square. — SteveKlinko

In order to consider the smallest possible square, we need some ontological principles, principles of physical existence which would dictate how small such an object could be. Otherwise it's just conceptual and there would be no limit to how small it could be. The same is the case for the largest possible square.

What is this Naturalism? How do you know Naturalism holds before the beginning. — SteveKlinko
Naturalism is the exclusion of magic from our consideration of the physical sciences.

I assert that ‘something from nothing’ is a magical proposition so we can exclude from our investigations of the origin of things. — Devans99

Yes I understand what you mean by Naturalism. But how can you know that Naturalism holds before the Beginning? Naturalism might only hold after a Universe comes into existence. But of course your Speculation could be correct. Any thing is possible related to knowing why the Universe is here.

But how can you know that Naturalism holds before the Beginning? — SteveKlinko

Science (or natural philosophy as it used to be called) is based on naturalistic explanations. Science, for example, excludes god and magic as valid explanation for natural phenomena.

If the early universe does not follow naturalistic rules then we have little hope of ever understanding it.

Rather than giving up, why not assume the universe behaves in a naturalistic ways and proceed to argue from there?

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.

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.