## Arguments in favour of finitism.

• 1.9k
Don’t assume your singular brain cell is infinite just to make yourself feel better.
• 305

I am a bit on the asperger side, but l can sense that you are kidding.
1/2 +1/4+1/8......till infinity =1
This converges.
Consider this series
-1+1-1+1.......infinity, this is not well defined.
So a bounded sequence can have a series that is not well defined.
• 305

:wink: there is no end to a joke about infinity.See what l did there.
• 856
No, I am not kidding.

Here's an example - take a picture, 100 by 100 pixels.
Infinitely zoom in - the picture is now infinitely large.
Infinitely zoom out - the picture is now infinitely small.
Each pixel within that picture is its own picture and has its own pixels.
In the end - it all converges in to one; one picture, one pixel.
• 324
What about infinite lines in euclidean geometry? Are they allowed? Are they the same thing as segments of length one?
• 135
Is there even such a thing as a strictly random number.

Perhaps we have no choice but to fix some meaning for 'random.' One fascinating way to do this is:

Kolmogorov randomness defines a string (usually of bits) as being random if and only if it is shorter than any computer program that can produce that string. To make this precise, a universal computer (or universal Turing machine) must be specified, so that "program" means a program for this universal machine. A random string in this sense is "incompressible" in that it is impossible to "compress" the string into a program whose length is shorter than the length of the string itself. — Wiki
https://en.wikipedia.org/wiki/Kolmogorov_complexity
• 856
They are. If you think otherwise, do please tell me why.
• 324
Which of the two questions do you answer? both?

So, you say that infinite lines in euclidean geometry are the same thing as segments of length one, right?

I thought this is obviously absurd. OK, if I have to say why: because infinite lines always intersect if they are not parallel, and finite segments can be not parallel and not intersect
• 856
Which of the two questions do you answer? both?
Quite.

I thought this is obviously absurd. OK, if I have to say why: because infinite lines always intersect if they are not parallel, and finite segments can be not parallel and not intersect
Yeah, I got that, but don't see the problem with it - if you look at infinite lines as stretched out finite lines.
You have two finite lines that are too short to intersect; you infinitely stretch them out and they intersect.
• 324
Yes, but we are speaking about measuring the length of segments. You can't stretch the segment if you want to measure it.
• 856
Sure you can.
To illustrate, you can infinitely divide a finite thing - which provides the notion it is an infinite thing, with infinite parts.

Now let's bring up the issue at hand.
Two finite things which are according to the aforementioned - likewise two infinite things, don't intersect.
But the idea is that two infinite things must intersect.
And my solution is simple - the halves of the one intersect in sum.
They are finite and likewise infinite - according to the aforementioned, and intersect either way you look at them.

And this comes about, due to it all always amounting to one.

Sure enough, you could just draw two lines that don't intersect and showcase that, but the two non-interlockers along with their void still amount to one.

Point being, if there is one, the lines can and cannot intersect, and all parts amount to one.
• 324
OK, so I can prove that 1 = 2. Here's the proof:

I take a segment of length one and I stretch it until it becomes of length 2. But this is still the same segment, so 1 = 2. What's the difference between this and your argument? The halves of two are the same as one. Why is your argument not valid for finite lengths?

I can even prove that a segment the same as a circle: just bend it and it becomes a circle!

If I want to compare the size of two objects I can operate on the objects only with transformations that don't change their size: for example I can put them one on top of the other by moving them, but I can't split them in two or stretch them.

I think I didn't understand completely your argument, but why can't you say the same thing for two instead of one?
• 324
if you don't allow the existence of infinite sets, you have to treat segments as a different kind of thing than a set of points.
But in the galois field, they treat the segment made of points but don't use infinite sets, the one mentioned in the article.Can you send me any article,book recommendations that make your position clear to me cause l may be confusing your point here, I hope not.

Sorry, I realize now that I didn't answer to this question.

I read the article that you posted (https://plato.stanford.edu/entries/geometry-finitism/supplement.html).
It's not true that "they treat the segment made of points". They say that "a point p corresponds to a couple (x,y)" and "A line corresponds to a triple (a,b,c)"; they don't say that "a line is a set of three points". But I agree that this is an example of finite models for Euclides' axioms, (by the way, in their original form Euclides' axioms are not expressed in a formal language, but I don't want to be picky on this point).

The infinite model that I was referring to is the standard one, based on the standard topology of the real line (https://en.wikipedia.org/wiki/Real_line).
I actually don't have a proof that there are no models of Euclides' axioms where a line is a finite set of points. On the contrary, I think you can easily build one by taking as space a finite 2-dimensional array of points, but this is obviously not the right model for the physical space.
So, probably I should have said that: if you want to build a model of the physical space where a segment is described as a set of points (as in standard topology), you need infinite sets. Otherwise, you have to build a model of the physical space where a segment is not a set of points (an example of this is kind, not based on set theory, is smooth infinitesimal analysis: https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis).
• 856
Quite. One halved is two, so by extension 1=2.

What I've said is valid for finite lengths, but if it sounds incoherent it's due my inability to properly explain. I'm not sure how to remedy that.

The best way I can put it is that one is the whole and thusly references all lengths without exception, and all lengths individually and as a contingent amount to one. The contingency of the whole amounting to repetition of one which derives distinctions precisely from the twists you mentioned.

I'm really not sure how to put it better, ironically, as it all amounts to how you put it. :)
• 461
Suppose somebody sends you a circle in the post. You then proceed to verify that the circle isn't perfect. Does "perfect" here refer to an internal property of the circle or to our inspection of it?

Here is what I think. Whenever we construct a set ourselves, we must choose the next element to include in the set, either explicitly, or by implicitly by defining a rule of selection. But in order to do so, it must be first be assumed that our elements are individuated a priori for a process of construction to make sense. Yet if we are given a set, say a parcel through the post, it's elements aren't individuated until we inspect the set. If the parcel we are given is called "infinite", all this means is that we shouldn't expect the termination of our parcel inspection to be decided by a property internal to the parcel.

Mathematics tends to call parcels with non-individuated elements "equivalence classes" of elements. Like in the above example, this allows mathematics to either construct sets in a 'bottom up' fashion from elements, or to construct elements in a 'top down' fashion from parcels.
• 324
Well, I see that you are not speaking of mathematics or logic here. So I think I have not much to say in reply to your answer...

But in my opinion the "infinity" in mathematics is not such an arcane entity as your "one" thing.
Infinitely big and infinitely small are the "simplification" of extremely big and extremely small.
So for example you can consider an infinitely long and infinitely thin line as a model of a very long and thin wire, because you want to "abstract away" the properties related to it's real length and width, and consider only the position and orientation of the wire.
The only reason that infinite numbers were ruled out from mathematics (on the contrary of infinite geometric objects, that were always considered allowed) was the idea that you cannot reason about them without getting inconsistencies in logic. But now we know how to deal with infinity using formal logic in a perfectly sound way, and nearly all important mathematics is making use of infinite and infinitesimals.

So, to me asking if infinite really exists is like asking if a plan really exists, since all "real" objects are 3-dimensional: it's only a mathematical approximation (simplification) due to the fact that you want to ignore one of the 3 dimensions. But probably this point of view is only due to my limited knowledge about philosophy..
• 1k
owever, the statement, " an infinite number of elements " is a contradiction in terms as infinity is clearly not a number

In the viewpoint that math is about symbol manipulation formalisms, we may not even be interested in the concept "meaning" as some kind of correspondence factor with the real, physical world.

For a starters, the symbol ∞ is just one character and not an ever-growing sequence of characters. Hence, it is perfectly suitable for participation in symbol-manipulation formalisms. An ever-growing sequence of characters, however, would be a problem, because in that case our symbol-manipulation algorithm may not even terminate.

Secondly, the symbol ∞ is not irrelevant or immaterial, because it is definitely mentioned in usable reduction rules that can successfully extend classical arithmetic. For example: a + ∞ = ∞ and also a * ∞ = ∞, with for example, a ∈ ℝ. The symbol clearly has some kind of "absorbing" effect on other elements in its domain. You can actually obtain/generate the ∞ symbol by performing particular manipulations on more common elements of the domain, such as 1/0 = ∞.

With the various reduction rules available, the symbol ∞ could actually be useful when you seek to produce a closed form output result from a particular input expression. The domain does not even need to be numerical and the algebraic structure not necessarily a field. The rule templates will undoubtedly still be consistent.

For example, with a and b arrays, elements from the Array domain, and a + b defined in a particular meaningful way, and ∞ a meaningful extension to the Array domain, you will find that a + ∞ = ∞.

So, it will still absorb the other elements during addition and multiplication. In that sense, it is a bit similar to the zero symbol, which also absorbs other elements after multiplication while leaving them unchanged after addition.

In other words, a field or other algebraic structure can successfully be extended with the symbol ∞ while maintaining consistency and while satisfying the pattern in existing rule templates for the symbol. So, infinity may indeed not be a number but it is certainly a legitimate extension element in numerical algebraic structures.
• 135
Secondly, the symbol ∞ is not irrelevant or immaterial, because it is definitely mentioned in usable reduction rules that can successfully extend classical arithmetic. For example: a + ∞ = ∞ and also a * ∞ = ∞, with for example, a ∈ ℝ. The symbol clearly has some kind of "absorbing" effect on other elements in its domain

Good point. It's even in floating point systems. I like the idea of an old cash register modified to ring up $\infty.$

I also like f($\infty$) for the limit of f as x goes to $\infty$. I find this kind of limit fairly intuitive. Many uses of infinity are intuitive in math. Finitism is cute, but it doesn't do justice to our intuition as a whole.

BTW, you mention math as pure reason. I relate to that formalist point of view, but I suggest that metaphor and intuition are important in doing math. I'd say that it's a kind of language.
• 305

:smile: I am late here.
In the viewpoint that math is about symbol manipulation formalisms, we may not even be interested in the concept "meaning" as some kind of correspondence factor with the real, physical world.
Let's consider the formalist view of math. I think that mathematics is primarily based on substitution where we replace a set of symbols with another set of symbols which are equal or equivalent in some cases. How do we decide that ? By "meaning" l meant the criterion for substituting one expression to another. Formalism has axioms and there are rules of inference etc. It cannot work without them.
For a starters, the symbol ∞ is just one character and not an ever-growing sequence of characters. Hence, it is perfectly suitable for participation in symbol-manipulation formalisms. An ever-growing sequence of characters, however, would be a problem, because in that case our symbol-manipulation algorithm may not even terminate.
The problem with using the infinity symbol is that there are infinities bigger than others. It is a single character but can we substitute it with numbers ? Consider the real line, all the real number lie on it but infinity doesn't. We can by some fancy definitions extend it to hyper-real and have the rules of adding numbers to infinity like a+infinity=infinity etc. Can you generate this symbol by any finite amount of operations ? I dont think we can and in my opinion formalism is basically about operations on symbols ? Therefore by allowing infinity, we sort of compromise the formal system. This is the basic idea behind the constructivist approach, if l am wrong, you are more than welcome to correct me.
With the various reduction rules available, the symbol ∞ could actually be useful when you seek to produce a closed form output result from a particular input expression. The domain does not even need to be numerical and the algebraic structure not necessarily a field. The rule templates will undoubtedly still be consistent.
I think that when we introduce the infinity symbol, we will have to drop associative law and commutative law too. There is a theorem by rieman which says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, or diverges. This doesn't apply to finite series for a reason and that is different laws regulate the symbols which have finite connotation and those which have an infinite connotation to them.
In other words, a field or other algebraic structure can successfully be extended with the symbol ∞ while maintaining consistency and while satisfying the pattern in existing rule templates for the symbol. So, infinity may indeed not be a number but it is certainly a legitimate extension element in numerical algebraic structures.
The real system has been extended to the hyper real and with it's own extended rules for operation but can we construct the equal or even equivalent of this symbol by same set of operation. By introducing the symbol into the rules and not being able to generate it from the real numbers is cheating. Is this extension valid ?
• 350
However, the statement, " an infinite number of elements " is a contradiction in terms as infinity is clearly not a number but sets have definite number of elements and on other hand infinity is not definite.How can we justify the existence of infinite sets.

Depends on how you define the word "number". If you define it narrowly, to mean "a natural number", then you're right, infinity is not a number. But that's not how most people define it. Likewise, if you define the word "set" narrowly, to mean "a finite collection of elements", then you're right, there are no infinite sets. But again, that's not how most people define it.

Infinity is a number and infinite sets are sets and that's all there is to it.

How can we justify the existence of infinite sets.

Make sure you don't confuse conceptual existence with empirical existence. The existence of the concept of infinite set is one thing and the existence of infinite collections of physical objects out there in the world is another. The former kind of existence is clearly real, the latter can be disputed.
• 305

Depends on how you define the word "number". If you define it narrowly, to mean "a natural number", then you're right, infinity is not a number. But that's not how most people define it. Likewise, if you define the word "set" narrowly, to mean "a finite collection of elements", then you're right, there are no infinite sets. But again, that's not how most people define it.
In case of sets, we use natural numbers to determine the cardinality but putting that aside, l would say there is 1-1 correspondence between all real numbers and a point on real number line, infinity doesn't lie there. How do you define numbers ? Most people do define a set in mathematics as a collection of well defined and distinct elements. Infinity is not an element even in the infinite natural set( or any other infinite sets). If you regard infinity as number, that implies that it is finite, since all numbers are finite.Hence a contradiction in terms.How do you define numbers ( the real numbers ) ? How can you justify infinity as a number ?
Make sure you don't confuse conceptual existence with empirical existence. The existence of the concept of infinite set is one thing and the existence of infinite collections of physical objects out there in the world is another. The former kind of existence is clearly real, the latter can be disputed.
I never compared the physical world with the mathematical one. Even if you consider the "conceptual existence " we cannot construct infinity even with all the symbols and operations in a system. This is not a physical limitation but a conceptual one. The concept of infinity does not allow it to be constructed out of numbers. Consider the case of halting problem precisely that of determining, from a description of an arbitrary computer program and an input, whether the program will finish running (i.e., halt) or continue to run forever. It was proven to be impossible and this is a conceptual restraint not an empirical one. Similar case applies to infinity.
Can you justify the infinity axiom.
• 1k
I think that mathematics is primarily based on substitution where we replace a set of symbols with another set of symbols which are equal or equivalent in some cases. How do we decide that ? By "meaning" l meant the criterion for substituting one expression to another.

If there is a match between a rewrite rule and an expression, you can rewrite the expression. So, it all depends on the rewrite rules of the system. These rewrite rules have no particular "meaning".

Say that there exists the following rewrite rule in the system: kk* --> k+, with k any arbitrary subexpression, then we can rewite the expression xyabc(abc)*rs --> xy(abc)+rs. This has no "meaning". The resulting expression is just the result of the mechanical application of the rewrite rule on the original expression.

Formalism has axioms and there are rules of inference etc. It cannot work without them.

Both axioms and rewrite rules are arbitrary. For example, the SKI combinator calculus uses the following rewrite rules:

Ix --> x. Kxy --> x. Sxyz --> xz(yz).

We can use this rewrite system to rewrite input expressions to output expressions, and derive new theorems from the system. For example, since SKxy -> y, we can see that SKx = I for any arbitrary choice of x. This theorem is meaningless, because the statement, which is provable from the construction logic of the SKI system, that "∀x in D: SKx = I" does not correspond to anything in the real, physical world.

It is a single character but can we substitute it with numbers ?

Infinity itself is a Platonic abstraction that is compatible with numbers, which are themselves also Platonic abstractions. Numbers are themselves no real-world objects either. Infinity is compatible when you can extend the rules for arithmetic to support the inclusion of infinity. while not damaging the algebraic structure.

Consider the real line, all the real number lie on it but infinity doesn't.

We don't care about lines in this context.

Therefore by allowing infinity, we sort of compromise the formal system. This is the basic idea behind the constructivist approach, if l am wrong, you are more than welcome to correct me.

I think that you correctly depict the constructivist view on infinity. But then again, I don't read much constructivist literature, because in my opinion, they are missing the point anyway. Cantor's elaboration of the concept of infinity is nicely consistent. I have no problem with it. I really do not see what the fuss is all about.

I think that when we introduce the infinity symbol, we will have to drop associative law and commutative law too.

Concerning commutativity, the problem does not seem to occur in x + ∞ versus ∞ + x, as both expressions get rewritten to ∞. In my impression, the arithmetic rewrite rules handle the symbol perfectly well. Adding ∞ should not modify the algebraic structure. Otherwise, you should not add it to the domain that you are dealing with. So, for example, if D' extends D with ∞, and <D,+> is a group, then <D',+> should also be a group. Otherwise, don't bother adding it.

By introducing the symbol into the rules and not being able to generate it from the real numbers is cheating. Is this extension valid ?

If adding it, can be done while preserving algebraic structure and therefore consistency, you can go ahead and add it. It does not even need to be about numbers.

In fact, there are situations where you must add ∞ in order to guarantee the consistency of field operations. For example:

For current cryptographic purposes, an elliptic curve is a plane curve over a finite field (rather than the real numbers) which consists of the points satisfying the equation y² = x³ + ax + b along with a distinguished point at infinity, denoted ∞. This set together with the group operation of elliptic curves is an abelian group, with the point at infinity as an identity element.

Without identity element, point addition would not be a group operation. The domain here does not consist of numbers but of two-tuples (x,y):

The equations (for point addition) are correct when neither point is the point at infinity, (0,0). When adding the point at infinity to another point, the result is simply the other point.

Elliptic curve arithmetic has obviously nothing to do with the real, physical world. It was not abstracted away from the real, physical world. Elliptic curve arithmetic is a Platonic abstraction that has characteristics and properties that turn out to be interesting, while adding a point at infinity is not only a requirement for consistency, but it also happens to work absolutely fine.
• 305
Say that there exists the following rewrite rule in the system: kk* --> k+, with k any arbitrary subexpression, then we can rewite the expression xyabc(abc)*rs --> xy(abc)+rs. This has no "meaning". The resulting expression is just the result of the mechanical application of the rewrite rule on the original expression.
If we look closely, you are in fact using k an arbitrary sub expression repeatedly. Let's say abcRefg where R is a relation or to simplify pRq, where the R can be an equal symbol or an inequality. Let's say we want to generate numbers from this system by a function F(x), where the input is a string of characters and the output is a number.
Can you generate infinity from that function ? I dont think so, unless you think we can construct an infinite number of characters in a string.
An absolute formal view will lead to many problems and there is also another problem with this language as it allows a function to take itself as an argument, that may lead to paradoxical self referential statements. Like a set of all sets for example.
Infinity itself is a Platonic abstraction that is compatible with numbers, which are themselves also Platonic abstractions. Numbers are themselves no real-world objects either. Infinity is compatible when you can extend the rules for arithmetic to support the inclusion of infinity. while not damaging the algebraic structure.
By alleging infinity to be a platonic abstraction doesn't help us understand its nature at all. I don't think there is a platonic world where all mathematical ideas can be found and the existence of non euclidean geometry proves that we have to create new maths a lot times by simply dropping some axioms ( parallel line axiom) and hence modify our system.
There is a transfer principle, to extend real to hyperreal and it is actually consistent as you mentioned in your earlier posts.
Elliptic curve arithmetic has obviously nothing to do with the real, physical world. It was not abstracted away from the real, physical world. Elliptic curve arithmetic is a Platonic abstraction that has characteristics and properties that turn out to be interesting, while adding a point at infinity is not only a requirement for consistency, but it also happens to work absolutely fine.
I just read about elliptic curves, the point at infinity isn't something lying at infinity. They use that term when they draw an elliptic graph on a 2d plane however in 3 dimensional geometry a line does intersect the elliptic curve at 3 points, case closed. Even, the real line can have a point at infinity by simply curling it around and making it meet at and end. This is in no way related to infinity as a concept. I think there is a misunderstanding here . I don't know much about elliptic curves but consider the aymtotoes of a hyperbola(x^2/a^2-y^2/b^2)=0 and then you get two aymtotoes and we say that they intersect the hyperbole at infinity, that does not mean they do. They keep getting closer and closer. You can never give the point of intersection there.
[0,1,0] is used as the point at infinity sometimes so that the equation p+q+r=0 is satisfied or some other form. When the need arises they let p+0=0 so l dont see how the point at infinity is related to infinity that we are discussing here. We also use the word intersect in different sense sometimes, for example if l say two parallel lines intersect each other at imaginary points, l am not referring to the normal case of intersection.
• 1k
Can you generate infinity from that function ? I dont think so, unless you think we can construct an infinite number of characters in a string.

Not sure. For example, if there is a rewrite rule "x/0 = ∞" for x not zero, the symbol could start popping up in output expressions. If you feed that output expression into your function F(x), it depends on whether F will accept it as an argument, and if so, if can successfully associate an output to it. Not sure at all.

You will need to apply two successive rewrites:

ab(ab)* -(1)-> ab+ -(2)-> cd+

(1) rewrite rule: KK*-->K+
(2) rewrite rule: ab-->cd

An absolute formal view will lead to many problems and there is also another problem with this language as it allows a function to take itself as an argument, that may lead to paradoxical self referential statements. Like a set of all sets for example.

Yes, Russell's paradox and Gödel's Incompleteness obviously apply. Axiomatic systems are quite powerful, but they also tend to be incomplete.

By alleging infinity to be a platonic abstraction doesn't help us understand its nature at all.

The nature of infinity is what you can do with it in your system. It will participate in rewrite rules. From there on, its nature emerges out of the rewrite rules in which it participates. If:

x & %SYMBOL = %SYMBOL

then %SYMBOL could have the role of the identity element, if domain D with operator "&" is meant to be a group. It could be part of the kernel of a homomorphism of sorts. The nature of %SYMBOL will become increasingly clear by considering the other rewrite rules.

When the need arises they let p+0=0 so l dont see how the point at infinity is related to infinity that we are discussing here.

It is just an example of an algebraic structure in which adding a concept of infinity keeps the entire system consistent. Otherwise, the domain has no identity element, and then is no longer a group. That is not allowed, because the system effectively makes use of the systemic property that the domain is an abelian group under addition. You can clearly see that in the definition of the encryption/decryption functions and of the sign/verify signature functions. When you prove that decryption is the inverse of encryption, you make use of the fact that it concerns an abelian group under addition.

This has nothing to do with the real, physical world. In this case, a point at infinity is just a tool to keep that cryptographical system consistent. This principle can be generalized. Adding infinity to a domain is just an instrument that can be used to achieve a particular purpose.

What is the constructivist alternative to doing that?

This approach is deeply embedded in existing technology nowadays:

The U.S. National Institute of Standards and Technology (NIST) has endorsed elliptic curve cryptography in its Suite B set of recommended algorithms, specifically elliptic curve Diffie–Hellman (ECDH) for key exchange and Elliptic Curve Digital Signature Algorithm (ECDSA) for digital signature. The U.S. National Security Agency (NSA) allows their use for protecting information classified up to top secret with 384-bit keys.

Therefore, it is a bit late in the game to argue whether extending a domain with the infinity symbol makes sense.
• 61
It's hard to even define infinity in a way that makes sense in reality, I think.

Mathematics and logic is one thing ... but reality is temporal and spacial. And if everything has always existed everywhere that just means since the beginning and in all places ... it doesn't actually imply infinity. The idea of infinite stuff seems rather odd, to me. There is however much there is. Everything is everywhere but there can't be more than there is ... and actual infinity would seem to suggest that there's more than there is... and that makes no sense.

The universe is eternal but finite, I say. Everything has always existed but that doesn't imply infinity because it doesn't imply before time because there is no time before time. There was a first moment—hence the universe is finite.
• 1k
Mathematics and logic is one thing ... but reality is temporal and spacial.

Only one part of the domain of knowledge, i.e. in Kantian lingo, "synthetic a posteriori", deals with the real, physical world. The other part, "synthetic a priori", only deals with abstract, Platonic worlds.

In my opinion, there may be too much emphasis on "reality".

In my professional activities, I have never dealt with "reality". I have only ever dealt with Platonic abstractions and their implementation in software. That is why I cannot comprehend why anybody would be so obsessed with the real, physical world to the exclusion of everything else.

Infinity is clearly a Platonic abstraction. Why would anybody try to shoehorn it into the real, physical world? For what purpose?
• 350
If you regard infinity as number, that implies that it is finite, since all numbers are finite.

That's according to your own definition of the word "number". You defined the word "number" to mean "a finite quantity". That's not the standard definition.
• 305

What is the standard definition of number ?
• 350
Even if you consider the "conceptual existence " we cannot construct infinity even with all the symbols and operations in a system.

The concept of infinity exists. That's what I meant by conceptual existence.
• 305

I hope you understand my point after reading this.
It explains my point.
Given that a mathematical extension is a symbol (‘sign’) or a finite concatenation of symbols extended in space, there is a categorical difference between mathematical intensions and (finite) mathematical extensions, from which it follows that “the mathematical infinite” resides only in recursive rules (i.e., intensions). An infinite mathematical extension (i.e., a completed, infinite mathematical extension) is a contradiction-in-terms.
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