What is the difference between actual infinity and potential infinity?

alcontali

And I don't understand the rest of your post, starting with "any language expression that matches only this kind of stuff, would be the membership function for a set of which the cardinality would be the powerset of real numbers" — GrandMinnow

A regular expression defines a regular language. For example, a* accepts {nothing, a, aa, aaa, aaaa, ...} or (ab)* accepts { nothing, ab, abab, ababab, abababab, ... } So, that what it accepts, is a set of sequences. The question is now: What is the cardinality of the set that it accepts? If it only accepts sets of sets of real numbers, then the set that it accepts is the power set of real numbers, and that would mean that its cardinality is beth2, i.e. 2^2^beth0. Is there a flaw in what I say?

GrandMinnow

As I understand, you're asking: What is cardinality of the set of finite sequences on a set of cardinality x?

I'm very rusty in math, so let me see if I remember correctly:

Let F(x) = the set of finite sequences on x.

Let w = the set of natural numbers.

If x is countable, then card(F(x)) = w. (That's for sure.)

If x is uncountable, then card(F(x)) = x? That follows by the absorption property of infinite cardinal arithmetic? (I'm almost sure that's right. But I'd have to check that I'm not overlooking a flaw in my premises.)

And card(PPR) = beth2.

GrandMinnow

Whether it's "true" in any meaningful sense is, frankly, doubtful. — fishfry

This is more a philosophical or psychological question than a purely mathematical one, but I don't have much problem understanding that the set of natural numbers and other infinite sets exist as abstract mathematical objects. And it's a pretty safe bet that mathematicians in general feel the same. Even as a child, I was introduced in a school textbook to the notions of the set of natural numbers, the set of rational numbers and the set of real numbers; and that did not seem problematic to me. And of course, as we know, whatever our feeling about the truth of real or abstract existence outside of the formal notion of existentially quantified theorems, no mathematical contradiction has been shown from ZFC.

TheMadFool

No, you just made the same mistake I pointed out the first time. — GrandMinnow

Let's go back to the beginning.

a = a + 1

1) a is NOTHING

infinity = infinity + 1

2) a is infinity

From 1 and 2, NOTHING = infinity by the reason that if it quacks like a duck, walks like a duck, it must be a duck

I don't know if that's a mathematical reason to say NOTHING = infinity but this is very simple logic. Kindly show me where I'm wrong.

Also I mentioned in my later post that NOTHING and infinity may not be the same thing despite similar behavior. Basically, I think NOTHING and infinity are points where math breaks down. Are you trying to say that?

TheMadFool

but I don't have much problem understanding that the set of natural numbers and other infinite sets exist as abstract mathematical objects — GrandMinnow

:brow: :chin:

alcontali

And card(PPR) = beth2. — GrandMinnow

Imagine what the regular expression accepts, are expressions like this:

{
{1.2323,343.3333}
,{344.2,0,34343.444,6454.6444}
,{2323.11,834.33}
,{}
,{5 12.1,99.343433}
}

So, it only accepts sets, the members of which must be sets themselves, and these member sets must only contain real numbers.

So, it only accepts elements from the power set of real numbers. (Correct?)

This regular expression seems to work like that (with members written in the alternative set notation):

\[\n((\d*(\.\d*)? ?)*\n?)*\]

So, I would like to confirm or infirm that :

card("\[\n((\d*(\.\d*)? ?)*\n?)*\]")=beth2

Metaphysician Undercover

It is not necessary to adopt platonism to accept that there are infinite sets. One may regard infinite sets as abstract mathematically objects, while one does not claim that abstract mathematical objects exist independently of consciousness of them. — GrandMinnow

There is a metaphysical problem with claiming that there are objects which do not exist independently of consciousness, and that is that these objects are imaginary. And imaginary objects are subjective, property of individual subjects. Such objects could be false, contradictory, or a logical impossibility. So if mathematical objects have this type of existence, each one needs to be justified, or else anyone could make up any imaginary thing, asserting that it exists as a mathematical object.

That is the problem with the infinite set. It is self-contradictory, an impossibility, which someone has asserted as an existing object, and other people have blindly accepted it because it is useful, without requesting justification. When we request justification, we see that "infinite set" is contradictory, as are most mathematical objects. And many which are not contradictory are irrational , like the principles of geometry.

Here's an example as to how mathematical objects are self-contradictory. Take the number 2. As an object, it is a simple unity. However, it is necessarily two distinct unities, as that's what 2 signifies, two distinct objects. So either 2 signifies two distinct things, or it signifies one unity, a mathematical object. It cannot signify both or else 2 would be 1, and that's contradictory. And so we cannot conceive of "mathematical objects" as objects, without loosing the meaning of the symbol. There is an inherent contradiction in asserting that a symbol like 2 signifies an object, because the unifying agent which makes 2 into one object has not been identified, therefore that two are one object has not been justified, and there really is no such object.

unenlightened

There is an inherent contradiction in asserting that a symbol like 2 signifies an object, because the unifying agent which makes 2 into one object has not been identified, therefore that two are one object has not been justified, and there really is no such object. — Metaphysician Undercover

I have a left shoe, and I have a right shoe.
I have two shoes.
I have a pair of shoes.

I have a pair of shoes, which consists of a left shoe, a right shoe and a unifying principle
I have a pair of shoes, which consists of a left shoe, a right shoe and two-ness.

Unfortunately, I have two left feet. This is not as bad as having two left brains.
--------------------------------------------------------------
I have two left shoes.
I do not have a pair of shoes.
I have a left shoe, and I have a left shoe, and I have two-ness, but not pairity.

Nansen Cuts the Cat in Two

Metaphysician Undercover

↪unenlightened

OK, you have two shoes. By what principle are these "two" things, one object? If they are a "pair" of shoes, this does not make them into an object, it is just another way of saying that they are two, a specialized form of "two". They are still not one object.

alcontali

When we request justification, we see that "infinite set" is contradictory, as are most mathematical objects. — Metaphysician Undercover

The concept of infinite set is abstract and very Platonic but not contradictory.

Furthermore, these different beth levels of infinite set sizes really kick in when you compare the set sizes of Platonic objects, which are obviously always language expressions in one way or another. Even natural languages are Platonic abstractions. For example, how many different sentences can you make in English? How does that compare to Chinese?

I tentatively guess the levels/beth numbers of infinity for English and Chinese will be the same. Still, in that case, how much is that beth level exactly, and how do natural languages compare to formal languages?

Another example. Context-free languages can in my impression express any arbitrary beth number because they can trivially handle wellformedness, while regular languages may only be able to reach level beth2 (not sure, though). That result would certainly be compatible with the Chomsky hierarchy of languages, in which regular languages are deemed substantially less powerful and expressive than context-free languages.

It is undoubtedly possible to prove a lot of these things, but I could not find any publication that deals with this matter.

Another result could be very interesting. If natural language has a particular fixed beth level, and since context-free languages can express any arbitrary beth level, there may exist context-free languages that are more powerful and more expressive than natural language. I don't know what that would mean, though.

Metaphysician Undercover

The concept of infinite set is abstract and very Platonic but not contradictory. — alcontali

It is contradictory, because a set is closed, complete, (as an object it is bounded, defined) whereas an infinity of anything is open, incomplete, unbounded and indefinite. I went through this in another thread recently, you weren't there.

alcontali

It is contradictory, because a set is closed, complete, (as an object it is bounded, defined) whereas an infinity of anything is open, incomplete, unbounded and indefinite. — Metaphysician Undercover

All possible sentences you can say in English is a set. How is that closed or complete? Has anybody ever been able to list these out? I don't think so.

unenlightened

If they are a "pair" of shoes, this does not make them into an object, it is just another way of saying that they are two — Metaphysician Undercover

I already shot that fox. If I have two shoes, they may or may not be a pair. It is not the same thing. And if I cut a cat in two, there are two pieces of one cat. Also not the same thing. But what do i have to do to make them one, tie the laces together - glue the soles together - crush them into a singularity?

Most of us know well enough how to count shoes and how to count pairs of shoes and bits of cat though, and we know well enough not to count the number as another shoe. So I am happy to say that however many shoes there may be, they are all shoes and not numbers, and though there is a number of shoes, there are only shoes and no numbers, and this is perfectly clear and simple until someone points out a contradiction, at which point the explanations multiply and the clarity is lost.

So don't do it.

alcontali

And if I cut a cat in two, there are two pieces of one cat. — unenlightened

Yes, but you are not allowed to put physical objects inside a mathematical set.
You can only fill it up with language expressions.
So, if you cut a "cat" in two, you get {"c", "at"} or {"ca","t"}.

If you want to put a real, physical cat inside a set, you need to do that in physics or so, or in one of the other real-world subjects. Math is language-about-language only. (furthermore, real-world disciplines use a completely different way of thinking about these things ...)

unenlightened

Yes, but you are not allowed to put physical objects into a mathematical set.
You can only fill it up with language expressions.
So, if you cut a "cat" in two, you get {"c", "at"} or {"ca","t"}. — alcontali

So are you saying that I cannot cut a cat into two pieces? Your idealism solves the problem of contradiction but at the price of failing to account for how we actually talk about the world. I don't talk about "cat"s but about cats - and sometimes I count them. Don't tell me I'm not allowed to... damn mathematicians and philosophers stealing the language the rest of us use to talk about the world and making silly rules against talking sense.

fishfry

This is more a philosophical or psychological question than a purely mathematical one, but I don't have much problem understanding that the set of natural numbers and other infinite sets exist as abstract mathematical objects. — GrandMinnow

I agree that mathematical infinity is "true" in the abstract realm of math. But that's like saying that the way the knight moves is "true" in chess. But it has no physical meaning in the world we live in. It's only true within a formal game played for entertainment. That's what I was trying to say.

fishfry

I wouldn't state it that way. If we mean first order Peano arithmetic (PA), then there are not in PA definitions of 'set', 'class', and 'proper class'. Meanwhile, in set theory, the domain of the standard model of PA is a set.
19 hours ago — GrandMinnow

I suppose so. But even in PA there are infinitely many numbers. There's just no completed set of them. So they are not formally a proper class, but we can use this idea as an analogy to what a proper class is. It's a collection that's too big to be a set. If I stated this as just a useful mental visualization or metaphor and not as a formal fact, we'd be in agreement I think.

alcontali

Your idealism solves the problem of contradiction but at the price of failing to account for how we actually talk about the world. — unenlightened

Any language is a Platonic abstraction that is mismatched with the real, physical world; even languages that are specifically meant to describe it.

With mathematics, the situation is even worse, because it is not even meant to describe the real, physical world, but only other language expressions. Mathematics is language about language. So, the real, physical world is out of scope in mathematics.

So, but yes, agreed, talking about the real, physical world, requires another regulatory framework that tries to keep the language expressions correspondence-theory "true", hopefully without degenerating into a complete mismatch.

Metaphysician Undercover

All possible sentences you can say in English is a set. — alcontali

No it isn't. A set consists of objects, not possible objects.

But what do i have to do to make them one, tie the laces together - glue the soles together - crush them into a singularity? — unenlightened

That's quite simple, to make them one, you have to refer to them as one, and not as two. if you refer to them as two things, a pair, or any such thing, then you are talking about two distinct things. But if you refer to them as one, then you are talking about one thing. But you cannot talk about them as two things and one thing at the same time without contradicting yourself. So either "two" refers to one object, a mathematical object, in which case it does not mean two distinct things, or "two" refers to two distinct things. You can use the word either way, but you must be careful not to equivocate, so you can't use it both ways at the same time.

So don't do it. — unenlightened

The problem is with the people who want to make mathematics do the impossible, not with the people who point out that what the mathematicians are doing when they're tying to make mathematics do the impossible, is contradictory.

alcontali

No it isn't. A set consists of objects, not possible objects. — Metaphysician Undercover

You do not need to populate a set with literal values. You can simply attach a indicator/membership function that is capable of letting through literals that belong to the set and keeping out literals that do not.

S1 = { 2, 3, 4, 5 }

S2 = { x | x ∈ N, x >=2 and x<=5 }

S1 and S2 describe the same set. Therefore, S1 = S2.

In mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of an element in a subset A of X, having the value 1 for all elements of A and the value 0 for all elements of X not in A. It is usually denoted by a symbol 1 or I, sometimes in boldface or blackboard boldface, with a subscript specifying the subset. In other contexts, such as computer science, this would more often be described as a boolean predicate function (to test set inclusion).

You can find another example for this principle in the definition for the term predicate:

Predicates are also commonly used to talk about the properties of objects, by defining the set of all objects that have some property in common. So, for example, when P is a predicate on X, one might sometimes say P is a property of X. Similarly, the notation P(x) is used to denote a sentence or statement P concerning the variable object x. The set defined by P(x) is written as {x | P(x)}, and is the set of objects for which P is true.

For instance, {x | x is a natural number less than 4} is the set {1,2,3}.

If t is an element of the set {x | P(x)}, then the statement P(t) is true.

Metaphysician Undercover

S1 and S2 describe the same set. Therefore, S1 = S2. — alcontali

That two things are equal does not mean that they are the same. This is a known deficiency of mathematics, equality cannot replicate identity. Anyone who argues that 2+2 is the same as 4 needs to learn the law of identity, and respect the difference between equality and identity. The two sets are not "the same" in the sense of "same" used in philosophy, they are "the same" in the sense of "same used by mathematicians (i.e. equal). In philosophy, an actual thing is not the same as a possible thing, and we have a law of identity to prevent this type of sophistry, employed by mathematicians who creep into philosophical discourse without the appropriate discipline.

alcontali

That two things are equal does not mean that they are the same. This is a known deficiency of mathematics, equality cannot replicate identity. Anyone who argues that 2+2 is the same as 4 needs to learn the law of identity, and respect the difference between equality and identity. — Metaphysician Undercover

Well, the one expression S1 consists of literals while the other expression S2 is a comprehension formula. So, they are indeed not identical but extensional, according to ZFC's axiom of extentionality.

In the axiom of extensionality, the "=" symbol has axiomatically been assigned to express extensionality. Hence, the conclusion that S1=S2 is accordance with the ZFC axiom.

The sentence "they both describe the same set and therefore they are extensional" is therefore in accordance with the axiomatic foundation of ZFC set theory.

The use of the "=" symbol for expressing extensionality can be confusing. In any programming language that I have ever run into, the expression "S1 = S2" (or usually "S1 == S2" ) will only compare the two data structures' memory storage addresses. If they happen to be stored in different locations, even if they contain the same elements, by default, they will be considered different. Potentially comparing each element would cost computing power, and that would not be desirable as a default interpretation, when consuming resources matters (like in computing but unlike in mathematics).

Metaphysician Undercover

The sentence "they both describe the same set and therefore they are extensional" is therefore in accordance with the axiomatic foundation of ZFC set theory. — alcontali

Right, but as I noted, this theory is deficient. The two sets are not the same set by any rigorous standard of "same", though they are "the same set" according to the deficient standard of ZFC set theory; the law of identity being the appropriate standard for "same", not ZFC theory. ZFC theory allows that two distinct things are the same, contrary to the law of identity. Since they are not the same, your argument, which requires that they are the same, to reach its conclusion, fails. Therefore they are not even equal.

You conclude that the two sets are equal based on the assumption that the two distinct descriptions describe "the same set". They do not describe the same set, by a rigorous standard of "same", therefore you cannot even conclude that the two sets are equal.

fishfry

ZFC theory allows that two distinct things are the same, contrary to the law of identity. — Metaphysician Undercover

I'm afraid I don't follow this at all. I know of no such instance.

The axiom of extensionality depends on the law of identity, which is a principle of logic and not of set theory. A thing is equal to itself. Then we define two sets to be equal if they have "the same" elements, meaning that we can pair off their respective elements using the law of identity.

It's true that in math we often identify sets as being the same type of "something" in a given context. For example the integers mod 4, the set {0, 1, 2, 3} with addition mod 4, are a very different set from the integer powers of the complex number i, {i, -1, -i, 1}. Yet the integers mod 4 (with the operation of addition mod 4) are isomorphic, as groups, to the powers of i under the operation of complex number multiplication. A group theorist will say these are "the same group" while being perfectly well aware that they're not the same set.

Another example is that if we define the natural numbers via the Peano axioms then use them to define the rationals and reals, then the natural number 1 and the real number 1 are entirely distinct sets. We regard them as the same via the "natural inclusion" of the naturals into the reals. If pressed on the details, any working mathematicians would explain this just as I have and there is never any confusion.

These issues are thoroughly discussed in a nice paper by Barry Mazur, "When is one thing equal to some other thing?"

http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf

I noticed that the example of 2 + 2 = 4 was given. 2 + 2 and 4 are exactly the same set. In the Peano axioms, 0 is a number and if n is a number, Sn, the successor of n, is a number. This gives an endless sequence 0, S0, SS0, SSS0, SSSS0, ... which we can use to name all the numbers. However this notation soon gets cumbersome so we adopt the definitions: S0 = 1, S2 = 2, S2 = 3, and so forth.

Now we can define addition inductively: n + 1 = Sn; and n + Sm = S(n + m). With this definition in hand, 2 + 2 = 4 is an easy theorem. If we then lift Peano to set theory via the axiom of infinity, 2 + 2 and 4 are easily seen to be the same set.

It's perfectly true that 2 + 2 and 4 are distinct strings of symbols. If we are studying strings, they're distinct. But in number theory they're the same number; and in set theory they're the same set.

alcontali

though they are "the same set" according to the deficient standard of ZFC set theory — Metaphysician Undercover

ZFC was initiated by Cantor and Dedekind in the 1870s, followed by Zermelo's draft 1908 publication, followed by Fränckel's bug fixes in 1921. From day number one, there has been forceful criticism on its choice of axioms, and there still is, with lots of people proposing alternatives. Still, ZFC's dominance has only kept growing.

Whatever happens, it will be really hard to replace ZFC by any alternative, because so many theorems now rest on it. ZFC has an enormous "installed base":

Installed base (also install base, install[ed] user base or just user base) is a measure of the number of units of a product or service that are actually in use, especially software or an Internet or computing platform,[1] as opposed to market share, which only reflects sales over a particular period. Although the install base number is often created using the number of units that have been sold within a particular period, it is not necessarily restricted to just systems, as it can also be products in general. For products which are in use on some machines for many years, the installed base count will be higher than sales over a given period. Some people see it as a more reliable indicator of a platform's usage rate.

ZFC is actually also a gigantic legacy system, without necessarily being outdated, though:

In computing, a legacy system is an old method, technology, computer system, or application program, "of, relating to, or being a previous or outdated computer system,"[1] yet still in use. Often referencing a system as "legacy" means that it paved the way for the standards that would follow it. This can also imply that the system is out of date or in need of replacement.

Having broken some teeth in the past by criticizing legacy systems with a large installed base while advocating their replacement, I now instinctively refrain from doing that, because the argument will most likely fail again. One reason is the Lindy effect:

The Lindy effect is a theory that the future life expectancy of some non-perishable things like a technology or an idea is proportional to their current age, so that every additional period of survival implies a longer remaining life expectancy. Where the Lindy effect applies, mortality rate decreases with time.

Bourbaki is also known to have strongly promoted ZFC.

Nicolas Bourbaki (French pronunciation: [nikɔla buʁbaki]) is the collective pseudonym of a group of (mainly French) mathematicians. Their aim is to reformulate mathematics on an extremely abstract and formal but self-contained basis in a series of books beginning in 1935. With the goal of grounding all of mathematics on set theory, the group strives for rigour and generality. Their work led to the discovery of several concepts and terminologies still used, and influenced modern branches of mathematics.

While there is no one person named Nicolas Bourbaki, the Bourbaki group, officially known as the "Association des collaborateurs de Nicolas Bourbaki" (Association of Collaborators of Nicolas Bourbaki), has an office at the École Normale Supérieure in Paris.

By the way, the "École Normale Supérieure" in Paris is also the school where Evariste Galois studied while working on his Galois Theory in his early twenties.

fishfry

ZFC was initiated by Cantor and Dedekind in the 1870s — alcontali

I agree with you re installed base or established mindshare. There are substantial developments in new foundations these days, category theory and homotopy type theory being the two leading candidates. In the end, foundations don't matter to the vast majority of working mathematicians. As an example if ZFC were found inconsistent tomorrow morning, it wouldn't affect group theorists or topologists or anyone else. They'd keep doing their work while the set theorists patched the problems. You'd be surprised how little attention working mathematicians pay to foundations.

alcontali

LOL. Impressive Wiki skills. No bearing on the topic at hand. What can I say? — fishfry

It is almost literally what you will find mentioned in the page on the "Brouwer-Hilbert controversy":

In other words: the role of innate feelings and tendencies (intuition) and observational experience (empiricism) in the choice of axioms will be removed except in the global sense – the "construction" had better work when put to the test: "only the theoretical system as a whole ... can be confronted with experience".

So, what happens with "the theoretical system as a whole" ? Either it finds downstream users, or else it doesn't. In that case, what can we say about the downstream use of ZFC? Well, it is a legacy system with an enormous installed base that has been around for almost a century. Does it matter? Well, according to the formalist philosophy, that is all that matters. The status of individual axioms is simply irrelevant.

Concerning "no bearing on the topic at hand", you undoubtedly say that, because you are not aware of that famous discussion between Hilbert and Weyl in 1927, which was exactly about this. Could that have something to do with "weaker" Wiki skills? ;-)

alcontali

These issues are thoroughly discussed in a nice paper by Barry Mazur, "When is one thing equal to some other thing?"
http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf — fishfry

The fact that the algebraic closures are not yoked together by a specified isomorphism is the source of some theoretical complications at times, while the fact that their automorphism groups are seen to be isomorphic via a cleanly specified isomorphism is the source of great theoretical clarity, and some profound number theory.

Yeah, the fact that automorphism groups are always isomorphic is a key ingredient in Galois theory. That is undoubtedly the "profound number theory" that he is referring to. ;-)

The Peano axiom approach calls up the full propositional apparatus of mathematics. But the details of the apparatus are kept in the shadows : you are required to “bring your own” propositional vocabulary if you wish to even begin to flesh out those axioms. The Peano category approach keeps all this in the dark: no mention whatsoever is made of propositional language.

Staying clear of the language of first-order logic may temporarily spare you from hitting the wall of Gödel's incompleteness theorems. It is the use of "∀" and "∃" that ransacks everything. Still, I do not see how he will manage to keep avoiding the use of language, and especially, existential quantifiers. Up till now, his Peano category approach has managed to somehow avoid their explicit use, but I am not sure that it also manages to avoid their implicit use.

The Peano axiom approach requires — at least explicitly — hardly any investment in some specific brand of set theory. At most one set is on the scene, the set of natural numbers itself. In contrast, the Peano category approach forces you to “bring your own set theory” to make sense of it.

Declaring the set theory in use to be some kind of free variable, or at least a template placeholder, is indeed interesting. However, how will he manage to not accidentally bring a particular set theory through the back door? All you need to do, is to accidentally rely on a theorem that rests on a particular set theory, in order to become beholden to it.

When we gauge the differences in various mathematical viewpoints, it is a good thing to contrast them not only by what equipment these viewpoints ultimately invoke to establish their stance, for ultimately they may very well require exactly the same things.

Ha ah! Exactly what I thought!

Representing one theory in another. If categories package entire mathematical theories, it is natural to imagine that we might find the shadow of one mathematical theory (as packaged by a category C) in another mathematical theory (as packaged by a category D). We might do this by establishing a “mapping”. We call such a “mapping” a functor from C to D.

Yes, I need a functor right now, between grammar classes of formal languages (which are always axiomatic theories). The PCRE regular language engine has custom extensions that allow it to express the grammar of context-free languages (EBNF) and match their sentences. So, now I want a functor between PCRE and EBNF; which are widely claimed to be isomorphic. So, does he know something about functors that would drastically simplify the job of producing such PCRE<-->EBNF functor? Otherwise, it may be a lot of work ... too much for me, in any case ...

Let X, X′ be objects in a category C. Suppose we are given an isomorphism of their associated functors η:FX∼=FX′. Then there is a unique isomorphism of the objects themselves,

Interesting and intriguing. Unfortunately, he does not mention the proof, even though he says it is an easy proof.

An object X of a category C is determined (always only up to canonical isomorphism, the recurrent theme of this article!) by the network of relationships that the object X has with all the other objects in C.

And you usually do not even need the object's relationship to ALL other objects. A few is usually enough to know what the object must be.

A functor F: C−−→D from the category C to D is called an equivalence of categories if there is a functor going the other way, G:D−−→C such that G·F is isomorphic to the identity functor from C to C, and F·G is isomorphic to the identity functor from D to D.

If anything that you can express in ZFC, can be expressed in combinatory logic, and the other way around, then there would be a equivalence functor between both categories. Then, this equivalence functor is also an algorithm, i.e. some kind of function that accept set-theoretical expressions and translates them in combinatory-logic ones. Has anybody ever implemented anything like that?

Is 5 mod 691 to be thought of as a symbol,or a stand-infor any number that has remainder 5 when divided by 691,or should we take the tack that it(i.e.,“5 mod 691”)is the (equivalence) class of all integers that are congruent to 5 mod 691?

Well, in my own experience, "5 mod 691" is just "5" in a system that happens to have as system parameter maxint=690. We do not really care about the system parameter particularly much, because everything we do, stays inside that system anyway. In my opinion, the choice of 691 would only matter when you simultaneously deal with multiple systems that could each have different parameters. Still, I have never run into that practical situation. Another reason why it does not matter, is because this system parameter will usually be relatively large. However, it will not be too large either, because the fact that numbers wrap around that maximum boundary has a desired obfuscating effect. It nicely ransacks monotonicity. So, 232+541 = 82 (within mod 691). So, you can perfectly add up two large numbers and get a smaller one. That is not a very strongly obfuscating effect, but it still helps in cryptography.

This newer vocabulary has phrases like canonical isomorphism,“unique up to unique isomorphism”, functor, equivalence of category and has something to say about every part of mathematics, including the definition of the natural numbers.

I also believe that category theory, i.e. general abstract nonsense, is the true flagship of mathematics. Unfortunately, its theorems do not (yet) have direct applications (such as in cryptography), that I know of.

The categorical vocabulary itself, however, seems to be spreading like wildfire.

Metaphysician Undercover

The axiom of extensionality depends on the law of identity, which is a principle of logic and not of set theory. A thing is equal to itself. Then we define two sets to be equal if they have "the same" elements, meaning that we can pair off their respective elements using the law of identity. — fishfry

By the law of identity, two distinct sets cannot be the same. If they actually are the same, then they are necessarily one, the same set. It's contradictory to say that two things are the same. If it is the same, it is only one. Being equal and being the same are very different because "equal" refers to a multitude while "same" according to the law of identity refers to one, and only one. If ZFC states that two equal things are the same, it clearly violates the law of identity, which necessitates that the appearance of two is an illusion, there is really just one (Leibniz principle). And we cannot talk about one being equal, because there is nothing for it to be equal with.

This is a fundamental problem with the so-called "objects" of mathematics. Distinct things are allowed to be the same object, contrary to the law of identity, through the means of a principle of equivalence. Mathematicians will defend the existence of these objects, as objects, through reference to a difference which doesn't make a difference. But strict adherence to the law of identity allows no such contradictory nonsense. If there is a difference between what "2+2" refers to, and what "4" refers to, then these cannot be the same object, despite the assertion that this is a difference which doesn't make a difference.

↪alcontali

I am concerned with the principles of the system, not any installed base, or legacy, these are irrelevant to the acceptability of the principles. I know that you believe axioms are completely arbitrary, making such things very relevant, so join the mob, if you like the "mob rules" philosophy.

alcontali

I am concerned with the principles of the system, not any installed base, or legacy, these are irrelevant to the acceptability of the principles. I know that you believe axioms are completely arbitrary, making such things very relevant, so join the mob, if you like the "mob rules" philosophy. — Metaphysician Undercover

There is the perennial requirement of consistency, but beyond that, anything flies, really. I am certainly very open-minded in mathematics. The more applications and users for such arbitrary concoction, the more likely that I will end up having a look at it. Arbitrary axioms are the hallmark of creativity! ;-)

What is the difference between actual infinity and potential infinity?

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