I think that it is not necessary for the infinite number of numbers to exist in my mind. All I need to have in my mind is S(n) = n+1. — Ludwig V

This is Aristotle's finitism. Finitism is like this: if we put you in a spaceship that has an odometer, you will never see any but a finite number on it. You'll never see an infinity symbol, though you never stop moving forever and ever.

Per Aristotle, infinity exists in potential. The actual is always finite. Set theory, by handling infinity as a set, appears to be defying finitism. This is an unresolved issue in phil of math. Someday it may result in a shift in thinking about set theory.

If I recall correctly, he specifically said "habit" rather than "rule", which suggests naturalizing logic, and indeed I think that's where he was headed. — Srap Tasmaner

There's a lot that could be said about that. It is tempting to classify some rules as habits. Habits neither have, nor require, any kind of justification. They are what they are and that's all that can be said. Rules, on the other hand, can provide justifications and, just for that reason, are, mostly subject to justifications. Some rules do not have, and do not require, any further justification - especially if they set the standard for right and wrong. So these are like habits in that they do not require justification and unlike habits in that they can provide justification. (I'm sketching here.)

We do not find numbers in nature, not in the same way we find trees and rocks and clouds. Did we then make it all up, our mathematics? Is it just a game we play with arbitrary rules? — Srap Tasmaner

That's true. But as soon as we recognize a tree, and recognise this is a different tree and this is the same tree again, we have have sown the seeds of counting. Not so much in the case of rocks and even less to in the case of clouds.
I don't think we make mathematics up like a story or even like a game (although the comparison can be useful). But it is a mistake to move from saying that we make mathematics up to saying that it (or its objects) don't exist. They don't exist in the way that trees and rocks do, but that shouldn't be a problem.
I don't like saying we discover or recognize mathematics either. It's not wrong, exactly, so long as we don't compare such discoveries to what explorers like Columbus do, thus positing a world of mathematics comparable to the physical world we live in.
Mathematics shares features with other practices in our lives and so comparisons can be useful. But I am unable to adopt any one comparison as the whole truth. (Comparison is always partial and involves differences as well as similarities.)
I'm in favour of naturalizing, if that means locating things in our lives and practices. But one has to acknowledge that there is a somewhat different project, at least in relation to mathematics, which is the search for foundations for a structure (as opposed to a set of practices).

If you think Meta has convincingly shown that numbers do not exist, then I suppose that's an end to this discussion. And to mathematics. — Banno

No, I don't think that @Meta has shown that numbers don't exist. I'm inclined to think that he doesn't believe that, either. He has been explicit that he rejects what he calls Platonism, but I don't think it follows that he thinks that numbers do not exist. I'm not sure he even rejects the idea that there are an infinite number of them - since he realizes that we can't complete a count of the natural numbers. I do think that we can't get to the bottom of what he thinks without taking on board the metaphysical theory that he has articulated.

I think that it is not necessary for the infinite number of numbers to exist in my mind. All I need to have in my mind is S(n) = n+1.
— Ludwig V
This is Aristotle's finitism. — frank

I didn't intend it to be. Surely, it works like this - Aristotle thinks that infinity cannot be real if we cannot complete the count. I intended to say that infinity is real even if we cannot complete the count, because the successor function tells us so.

Per Aristotle, infinity exists in potential. The actual is always finite. Set theory, by handling infinity as a set, appears to be defying finitism. This is an unresolved issue in phil of math. Someday it may result in a shift in thinking about set theory. — frank

Yes, I'm aware of that - and of the startling results that followed when his view was set aside and infinity was treated as real, thus enabling the invention/discover/development of the calculus.
Come to think of it, one might argue that Aristotle does not actually say that infinity doesn't exist, just that it exists ("in potential"). But everyone thinks that something that potentially exists doesn't exist and it would be perverse to suggest otherwise, I suppose.

Yes, I'm aware of that - and of the startling results that followed when his view was set aside and infinity was treated as real, thus enabling the invention/discover/development of the calculus. — Ludwig V

Aristotle is not set aside by calculus because it does not deal with actual infinity. Set theory is a different matter.

So you do know that the series is infinite without completing the count of them all. — Ludwig V

Of course, why would say that? it's defined as infinite. That's the whole point. It is infinite and infinite is defined as boundless, endless, therefore not possible to count. So any axiom which states that it is countable contradicts this.

And yet, Frodo Baggins exists - in the way that fictional characters exist. They can even be counted. Similarly, numbers exist - in their way. — Ludwig V

Yes, a finite number of fictional thigs is countable. But infinite is defined as endless, therefore it is impossible to count an infinite number of fictional things.

I'm not quite sure that I understand you. I think that it is not necessary for the infinite number of numbers to exist in my mind. — Ludwig V

I'll explain again. If numbers are assumed to be independent Platonic objects, we can assume that bijections simply exist, without needing to be produced by a human being. However, the infinite bijection is a matter of contradiction, even if bijections simply exist. Therefore it ought to be rejected as incoherent. If, on the other hand, numbers are assumed to be fictional objects, created by human minds, the same contradiction still remains. It is the idea that numbers are infinite, yet countable as objects, which is incoherent. So it does not matter how you validate the existence of numbers as objects which can be counted, the incoherency cannot be avoided.

All I need to have in my mind is S(n) = n+1. — Ludwig V

I don't understand why, so many people on this thread seem to think that if they can make symbols which represent something incoherent, this somehow makes it coherent. When we speak contradictions, that's what we do, use symbols to represent something incoherent. Why would you think that writing it somehow makes it coherent? I can say RNR stands for the thig which is both red and not red at the same time, but how does symbolizing it make it coherent? How does "S(n) = n+1" make you believe that an uncountable number of objects is countable? I truly cannot understand this.

It turns out that the disagreement turns on a metaphysical disagreement. Tackling that needs a different approach. — Ludwig V

It's not a metaphysical problem directly. As a matter of contradiction between basic axioms, it is an epistemic problem within the mathematical system (set theory). As explained above, it doesn't matter which metaphysics you use to validate the existence of numbers as countable objects, the problem remains.

I do propose that it could be resolved with a metaphysical solution though. The solution is to reject the ontology which supports the idea that numbers are countable things, along with the mathematics which follows (set theory). An idea is not a thing which can be counted, and that is a basic flaw in the ontology which supports set theory.

Notice, it's not a metaphysical problem in itself. We can assume that numbers and all sorts of ideas are objects, and maintain that ontology. The problem is epistemic. We think that since numbers are objects then they ought to be countable. that's what produces this problem. To resolve the problem we might say that numbers are a type of object which is for some reason not countable, but that creates a problem with the concept of "object". Therefore it's better, and actually provides a better foundation for understanding what concepts are, if we deny that numbers are objects.

If you think Meta has convincingly shown that numbers do not exist, then I suppose that's an end to this discussion. And to mathematics.

But I hope you see the incoherence of his position. — Banno

The point is that a number is not a thing which can be counted, it is something in the mind, mental. I think you understand the difference between physical, sensible things which can be counted, and mental thoughts which are not individual things that might be counted. You did read Wittgenstein's Philosophical Investigations didn't you? Did you learn anything from it?

There is a very significant error in the idea that a measuring system could measure itself.

No, I don't think that Meta has shown that numbers don't exist. I'm inclined to think that he doesn't believe that, either. He has been explicit that he rejects what he calls Platonism, but I don't think it follows that he thinks that numbers do not exist. I'm not sure he even rejects the idea that there are an infinite number of them - since he realizes that we can't complete a count of the natural numbers. I do think that we can't get to the bottom of what he thinks without taking on board the metaphysical theory that he has articulated. — Ludwig V

The point is that "numbers" do not exist as individual countable things. This is a misrepresentation of what a number is, and the problem becomes evident when we allow the infinite capacity of numbering, and then try to count those numbers. So it doesn't matter if you represent the number as an independent Platonic object, or an object of human construct, either way is faulty. A supposed individual number is really an idea, which is dependent on other ideas for its meaning, and cannot be accurately represented as an individual object.

it's defined as infinite. — Metaphysician Undercover

Maybe for you. For me, that's a theorem.

The point is that a number is not a thing which can be counted — Metaphysician Undercover

There is a very significant error in the idea that a measuring system could measure itself. — Metaphysician Undercover

Then this is nothing to do with infinite sequences, infinite sets, or infinity.

Your position is that you can't count how many numbers there are between 1 and 10.

The point is that a number is not a thing which can be counted, it is something in the mind, mental. — Metaphysician Undercover

Now many integers are there between zero and five?

Of course, why would say that? it's defined as infinite. That's the whole point. It is infinite and infinite is defined as boundless, endless, therefore not possible to count. So any axiom which states that it is countable contradicts this. — Metaphysician Undercover

"Infinite" means "not finite", not "not countable".

And to be countable is to have an injection into ℕ; or equivalently, to have a bijection with a subset of ℕ. For infinite items, that subset is ℕ itself.

ℕ is both infinite and countable. ℝ is infinite and uncountable.

You have yet to show a contradiction in bijection; indeed, you have yet to show what that might even mean. Mathematics on the other hand takes a bijection between two sets A and
B to mean there is a rule f such that each element of A is paired with exactly one element of B, and each element of B is paired with exactly one element of A. Here is a proof that ℕ has a bijection with ℕ, and so is countable:

Take
$f : \mathbb{N} \to \mathbb{N}, \quad f(n) = n$

• Well-defined:
  For every $n \in \mathbb{N}$, we have $f(n) = n \in \mathbb{N}$.
  Hence $f$ is well-defined.
• Injective:
  Suppose $f(n_1) = f(n_2)$. Then
  $n_1 = n_2$, so $f$ is injective.
• Surjective:
  Let $m \in \mathbb{N}$. Define $n = m \in \mathbb{N}$. Then
  $f(n) = f(m) = m$, so every element of $\mathbb{N}$ is hit by $f$.
  Hence $f$ is surjective.

Conclusion:
The function $f(n) = n$ is a bijection from $\mathbb{N}$ to $\mathbb{N}$.
Therefore $\mathbb{N}$ is countable.

The bijection is not assumed, it is demonstrated.

I can't really follow anything you are saying.

Now many integers are there between zero and five? — Banno

Again, "integer" is a faulty concept, because it assumes that "a number" is a countable object. That's exactly the problem I explained to you. We ought not treat an idea as an individual object. Providing more examples of the same problem will not prove that the problem does not exist. The problem of Platonism is everywhere in western society, even outside of mathematics, so the examples of it are endless.

Mathematics on the other hand takes a bijection between two sets A and
B to mean there is a rule f such that each element of A is paired with exactly one element of B, and each element of B is paired with exactly one element of A. — Banno

This does not address the point. A rule can contradict another rule within the same system. Saying that there is a rule which allows a specific bijection doesn't necessarily mean that there is not another rule which disallows such.

The bijection is not assumed, it is demonstrated. — Banno

That's false, it's not a demonstration, at best, it's begging the question. You have no definition of "countable", so your conclusion, "Therefore N is countable" does not follow. It would follow, if you provide a definition of "countable" which begs the question. The proper conclusion is N is countable if N is countable. Then a definition of "countable" could be provided which contradicts the infinite nature of the natural numbers, making "N" "countable". Voila, begging the question with contradiction. I think Magnus already explained this to you, so you're just continuing to demonstrate your dishonest denial. I don't see any point to further discussion, you'll only continue to refuse to look at what is shown to you, and rehash the same faulty arguments.

Again, "integer" is a faulty concept, because it assumes that "a number" is a countable object. — Metaphysician Undercover

So apples are countable, but numbers aren't. :grin:

You got it bro!
We measure the object not the measurement tool. The standard metre cannot be measured. Numbers are the tool, not the thing to be measured.

So apples are countable, but numbers aren't. — frank

:joke:

Oh oh, the set {1,2,3} has 3 numbers. :gasp:

It's called nominalism. I would ask one favor though. Stop capitalizing the P in Platonism. The phil of math view of platonism. Plato pitted opposing ideas against each other, so for instance, in Parmenides, he outlines a lethal argument against the Forms. That's why they use a little p: platonism.

1. Two views about mathematics: nominalism and platonism

In ontological discussions about mathematics, two views are prominent. According to platonism, mathematical objects (as well as mathematical relations and structures) exist and are abstract; that is, they are not located in space and time and have no causal connection with us. Although this characterization of abstract objects is purely negative—indicating what such objects are not—in the context of mathematics it captures the crucial features the objects in questions are supposed to have. According to nominalism, mathematical objects (including, henceforth, mathematical relations and structures) do not exist, or at least they need not be taken to exist for us to make sense of mathematics. So, it is the nominalist's burden to show how to interpret mathematics without the commitment to the existence of mathematical objects. This is, in fact, a key feature of nominalism: those who defend the view need to show that it is possible to yield at least as much explanatory work as the platonist obtains, but invoking a meager ontology. To achieve that, nominalists in the philosophy of mathematics forge interconnections with metaphysics (whether mathematical objects do exist), epistemology (what kind of knowledge of these entities we have), and philosophy of science (how to make sense of the successful application of mathematics in science without being committed to the existence of mathematical entities). These interconnections are one of the sources of the variety of nominalist views. — SEP

Oh oh, the set {1,2,3} has 3 numbers. :gasp: — jgill

A nominalist entirely rejects set theory because it's a mountain of abstract objects.

I would ask one favor though. Stop capitalizing the P in Platonism. The phil of math view of platonism. Plato pitted opposing ideas against each other, so for instance, in Parmenides, he outlines a lethal argument against the Forms. That's why they use a little p: platonism. — frank

I agree. but my spell check doesn't like little p platonism. And, I count the distinction as unimportant because there really would be no such thing as big P Platonism if we maintained that distinction. Plato pitted ideas against each other so there's no real ontological position which could qualify as big P Platonism. So they end up being the same meaning anyway.

I agree. but my spell check doesn't like little p platonism. And, I count the distinction as unimportant because there really would be no such thing as big P Platonism if we maintained that distinction. Plato pitted ideas against each other so there's no real ontological position which could qualify as big P Platonism. So they end up being the same meaning anyway. — Metaphysician Undercover

I think you're discounting the importance of community. If it's not stretching your spine out of shape, you can go along with the rest of the phil of math and write it as platonism. It's a little nod to the deep bonds that hold us together over the millennia as our brothers and sisters try to take freakin' Greenland and what not.

Again, "integer" is a faulty concept, because it assumes that "a number" is a countable object. That's exactly the problem I explained to you. We ought not treat an idea as an individual object. Providing more examples of the same problem will not prove that the problem does not exist. The problem of Platonism is everywhere in western society, even outside of mathematics, so the examples of it are endless. — Metaphysician Undercover

We can make it simpler for you: How many whole numbers are there between one and three?

I say one. You say, they can't be counted.

A rule can contradict another rule within the same system. Saying that there is a rule which allows a specific bijection doesn't necessarily mean that there is not another rule which disallows such. — Metaphysician Undercover

Then show us that other rule. Pretty simple. Set the supposed contradiction out.

That's false... — Metaphysician Undercover

“Countable” is defined as “there exists a bijection with ℕ (or a subset of ℕ).” I bolded it for you. Again, if you think there is a contradiction in that, it is up to you to show it.

Begging the question would look like this:

• Assume N is countable.
• Therefore
• N is countable.

But what actually happens is:

• Definition: “countable” = “there exists a bijection with N”.
• Construction: Exhibit such a bijection (the identity).
• Conclusion: Therefore N is countable

.
That is definition + witness, not circularity.

You of course do not have to respond to my posts. Keep in mind that everything I have set out here is standard ZFC, and has been examined by countless mathematicians, yet remains solid. It is your account that is eccentric.

Yep.

I think you're discounting the importance of community. If it's not stretching your spine out of shape, you can go along with the rest of the phil of math and write it as platonism. It's a little nod to the deep bonds that hold us together over the millennia as our brothers and sisters try to take freakin' Greenland and what not. — frank

This issue is more complicated though. The Neo-Platonists took Plato's name and claimed to continue Plato's school, but their ontology is consistent with what you call platonist. Aristotle's school claimed to be the true Platonists but the Neo-Platonists took the name. So you have to take on the Neo-Platonists, and tell them that they should call themselves Neo-platonists, as not true Platonists. But this problem has been around for millennia, and they do not like being accused of misrepresenting Plato, they like to claim the true continuation of Plato's teaching.

We can make it simpler for you: How many whole numbers are there between one and three? — Banno

i say it's a loaded question, like "have you stopped beating your wife?". If we give up on the idea that there are numbers in between numbers, we get rid of an infinity of problems from infinitely trying to put more numbers between numbers. This supposition that you have, that there are numbers between numbers is very problematic.

Set the supposed contradiction out. — Banno

I did it all ready in this thread, numerous times. If you're truly interested go back and reread my posts. But I'm tired of it. And I know you, you'll just deny anyway so what's the point?

“Countable” is defined as “there exists a bijection with ℕ (or a subset of ℕ).” I bolded it for you — Banno

Right, begging the question. "There exists a bijection with N" is explicitly saying "N is countable". Are you kidding me in pretending that you don't see this?

This issue is more complicated though. The Neo-Platonists took Plato's name and claimed to continue Plato's school, but their ontology is consistent with what you call platonist. Aristotle's school claimed to be the true Platonists but the Neo-Platonists took the name. So you have to take on the Neo-Platonists, and tell them that they should call themselves Neo-platonists, as not true Platonists. But this problem has been around for millennia, and they do not like being accused of misrepresenting Plato, they like to claim the true continuation of Plato's teaching. — Metaphysician Undercover

I am a Neoplatonist, and I don't care whether you capitalize the P or not! :grin:

This supposition that you have, that there are numbers between numbers is very problematic. — Metaphysician Undercover

So your argument is that 2 is not between 1 and three.

Righto.

I did it all ready in this thread, numerous times. — Metaphysician Undercover

Well, no. You claimed there is a contradiction, repeatedly, but never showed what it was. So go ahead and quote yourself.

"There exists a bijection with N" is explicitly saying "N is countable". Are you kidding me in pretending that you don't see this? — Metaphysician Undercover

"There exists a bijection of N" is the conclusion, not an assumption.

So your argument is that 2 is not between 1 and three. — Banno

He's saying that 2 isn't a thing. It's a modifier like pink. You can't count pinks because it's not a thing you count. It's nominalism.

He's saying that 2 isn't a thing. — frank

Yes, indeed he is.

And the counterpoint is that "being a thing", especially in mathematics, consists in being the value of a bound variable. As in "for any whole number, if it is between one and three then it is two"

This is not saying numbers are spooky abstract objects. It's saying that whatever our theory quantifies over, exists according to that theory. The nominalist’s complaint about “thinghood” simply misses the target.

It's a modifier like pink. You can't count pinks because it's not a thing you count. It's nominalism. — frank

“Pink” is not something that can be the value of a bound variable in arithmetic. “2” is. The analogy fails.

The other error here is to think that If something is countable, it must be a concrete or abstract thing. it ain't so. And note the equivocation of "thing".

I don't think it's that simple. It really comes down to which view best accommodates what we do with math.

It really comes down to which view best accommodates what we do with math. — frank

Ok. Here's some stuff that won't work if we accept Meta's ideas.

• Quantification
  If numbers are not admissible as values of bound variables, then statements like “for any natural number n” or “there exists a number such that” are illegitimate. This eliminates axiomatic arithmetic, algebraic generality, and proof by universal or existential instantiation.
• Identity and equality
  Arithmetic relies on identity conditions such as 2 = 2, 2 ≠ 3, and “if n is between 1 and 3, then n = 2”. If numbers are merely modifiers, they cannot enter identity statements, cannot be uniquely satisfiable, and cannot ground equality.
• Ordering relations
  Relations like less than, greater than, and between require relata. If numbers are not entities in any sense, then statements like “2 is between 1 and 3” are not well-formed, and order theory collapses.
• Counting finite collections
  Even finite arithmetic fails. Claims such as “there is exactly one whole number between 1 and 3” or “this set has three elements” require individuation, discreteness, and cardinality. These cannot be recovered without smuggling in what the view denies.
• Functions
  Functions are mappings (e.g. f : ℕ → ℕ). If numbers are not admissible values, then functions have no domain or codomain, expressions like f(2) are meaningless, and recursion is impossible.
• Proof by construction
  Mathematics routinely proves existence by exhibiting a value (“let n = 2”). If numbers cannot be introduced as values, constructive proofs and witness-based reasoning disappear.
• Set theory
  Set theory quantifies over elements (e.g. 2 ∈ {1,2,3}). If numbers are not legitimate elements, sets of numbers are incoherent, cardinality is undefined, and bijections cannot be stated.
• Algebraic structure
  Even structuralism requires positions in structures. If individuation is denied altogether, then groups have no elements, rings have no units, and fields have no values. Structure without positions is empty.
• Application of mathematics
  Physics, engineering, and statistics require numerical values, parameters, and measurements. Treating numbers as mere modifiers strips equations of semantic content and collapses measurement and prediction.
• Self-undermining practice
  The view relies on finite counting, numerical distinction, and identity (“one”, “two”, “numerous times”) in order to be stated at all. It presupposes the very arithmetic it rejects.

We could go on.

Yes. Nominalists believe we don't need to posit abstract objects to make sense of math. It's generally considered that they have the burden of proof, and they take that seriously.

Sure. But in addition to the usual thngs nominalism rejects, Meta rejects the notion that numbers as values of variables. while nominalists say numbers aren’t abstract objects, they undersntad that they can still be quantified over. Meta says that numbers aren’t things at all — they’re modifiers like “pink”. That blocks:

• ∀n …
• ∃n …
• n = 2
• n < 3

No mainstream nominalism does this, because it destroys the grammar of mathematics.

And so on. It's not nominalism as usually understood. Even predicativist or fictionalist views preserve quantificational structure.

Ok. I'm probably wrong then.

I've tried to follow what you are doing here, but scattered inaccuracies and errors make it very difficult. I gather you want to Cantor’s argument into a constructive or even computational lens. It’s valid in that framework, yet you seem to think it can be taken as refuting classical results about cardinality. — Banno

Yes absolutely, if we interpret "refuting cardinal analysis" as ditching ZF/ZFC for being computationally inadmissible due to the infinite hierarchy of cardinals being computationally meaningless and poorly motivated, given the fact that ZF/ZFC are set theories that are descendents of Cantor's theological prejudices that aren't true by correspondence to anything of relevance to science and engineering.

More specifically, the Cantor-Schröder-Bernstein Theorem that is the foundation of infinite cardinal analysis, is abjectly false in any constructive intepretation of the diagonalization argument that is conscious of undecidability.

In reverse mathematics, where we start by analysing a theorem without first assuming a particular axiomatic foundation, then the CSB theorem becomes the assumption that if f : A --> B is an injection (written |A| <= |B|) and g : B --> A is an injection (written |B| <= |A|) , then there must exist a bijection between A and B (written |A| = |B|).

So let's take P(N) to be the decidable subsets of the natural numbers. Then is CSB true or false?

1. We know that we can construct an injection P(N) --> N via Turing machine encoding of decidable sets. (|P(N)| <= N)

2. We can build 'any old' injection f : N -> P(N) to show that |N| <= |P(N)|.

3. Hence according to CSB, the set of decidable sets of P(N) has the same size as N.

And yet f cannot be a surjection: For diagonalising over f must produce a new member of P(N), but this isn't possible if f is surjective. Hence f cannot be a surjection, and this is the reason why diagonalization can produce new members of P(N), without P(N) ever being greater than N.

This particular case turns CSB against it's originator Cantor, for CSB insists that P(N) and N must necessarily have the size, in spite of the fact a surjection N --> P(N) cannot exist.


So this is where Cantor specifically went wrong: he should have interpreted diagonalization as showing that a surjection cannot always exist between countable sets. But instead, Cantor started with the premise that a surjection A --> B must always exist when A and B are countable, which forces the conclusion that diagonalisation implies "even bigger" uncountable sets, which is a conclusion that Cantor accepted because it resonated with his theology.

So your argument is that 2 is not between 1 and three. — Banno

That's exactly right. To say that 2 is between 1 and 3 is to say that it serves as a medium. However, in the true conception and use of numbers, 1, 2, 3, is conceived as a unified, continuous idea. This unity is what allows for the simple succession representation which you like to bring up. No number is between any other number, they are conceived as a continuous succession. To say that 2 comes between 1 and 3 is a statement of division, rather than the true representation of 1, 2, 3, as a unity, in the way that the unified numbering system is conceived and applied.

Well, no. You claimed there is a contradiction, repeatedly, but never showed what it was. So go ahead and quote yourself. — Banno

"Infinite" means limitless, boundless. The natural numbers are defined as infinite, endless. limitless. All measurement is base on boundaries. To say a specific parameter is infinite, means that it cannot be measured. Counting is a form of measurement. Therefore the natural numbers cannot be counted. To propose that they are countable, is contradictory, because to count them requires a boundary which is lacking, by definition.

This is why "open sets" are used to justify unmeasurable spaces, resulting in an incoherent concept of continuity. Incoherent "continuity" is the result of the false opinion that there exists numbers "between" numbers, instead of representing the numbers as a unified concept.

Look, to say that something is infinitely heavy or light means its weight cannot be measured. To say that it is infinitely long or short means that its length cannot be measured. To say that it's infinitely hot or cold means that its temperature cannot be measured. To say that it is an infinitely large or small quantity means that its quantity cannot be measured.

Why do you think the proposition that the natural numbers is countable does not contradict the proposition that the natural numbers are infinite, in the way I explained?

But in addition to the usual thngs nominalism rejects, Meta rejects the notion that numbers as values of variables. while nominalists say numbers aren’t abstract objects, they undersntad that they can still be quantified over. Meta says that numbers aren’t things at all — they’re modifiers like “pink”. That blocks: — Banno

Correction. There is no difference between a number and a numeral, the number is the symbol. What the symbol represents is an abstract value. It's a category mistake to say that a value is an "object" unless we define "object" in the sense of a goal.

Regardless of what you assert, to say that the value represented is an object called "a number" is platonism. Calling it an imaginary, or fictional object, doesn't fulfill the ontological criteria of "object". Therefore we'd have to treat it as an idea because treating it as an object would be a false premise. We cannot truthfully treat a fictitious object as an object, because it is an idea and the existence of ideas is categorically distinct from the existence of objects.

Regardless of what you assert, to say that the value represented is an object called "a number" is platonism. — Metaphysician Undercover

:heart:

"Infinite" means limitless, boundless. The natural numbers are defined as infinite, endless. limitless. All measurement is base on boundaries. To say a specific parameter is infinite, means that it cannot be measured. — Metaphysician Undercover

As a slogan, that looks almost right.

To say that a parameter is infinite, means that it cannot be measured relative to a given basis of description. Hence the distinction between an ordinary task and a hypertask depends on how the task is described, and this distinction can be regarded as the logically correct solution to Zeno's Dichotomy paradox.

(The finitude of an object's exact position in position space, becomes infinite when described in momentum space, and vice versa. Zeno's paradox is dissolved by giving up the assumption that either position space or momentum space is primal)