'inherent' has not been given a mathematical definition. Dispute about it can go on and on and on, and in circles, for as long as people have the opportunity and willingness to dispute about it.

But notions that have been given mathematical definition include:

ordinal
well ordering
ordinal less-than
least ordinal in a set
least ordinal with a formula defined property
bijection
equinumerous
isomorphism
order type
injection
cardinal
cardinality
cardinal less-than

Definitions adhere to forms that ensure ensure eliminability (formulas with defined terms can be set back to formulas without the defined terms) and non-creativity (formulas that weren't already provable aren't made provable with the introduction of defined terms). By adherence to the forms for definitions, the definitions are never circular.

And new definitions can be provided - anyone is free to introduce a new term and define it.

Set theory is not properly critiqued by acting as if some undefined terms controls results in set theory.

df: S is equinumerous with T <-> there is a bijection between S and T

theorem: For every S, there is a unique T such that T is an ordinal & S and T are equinumerous & no member of T is equinumerous with S

df: card(S) = the unique T such that T is an ordinal & S and T are equinumerous & no member of T is equinumerous with S.

We say card(S) is the cardinality of S.

df: k is ord-less-than j <-> k e j

When context is clear, we just say "k is less than j" or "k < j" or "k e j".

df: k is ord-least in S <-> (k e S & ~Ej(j e S & j ek))

When context is clear, we just say "k is least in S".

df: k is the least ordinal such that P <-> (k is an ordinal & Pk & ~Ej(Pj & j ek))

df: S is a cardinal <-> (S is an ordinal & there is no ordinal T less than S such S and T are equinumerous)

theorem: S is a cardinal <-> Ex S = card(x)

/

There is no circularity there.

If one has a definition of 'inherent' then they can add it.

I answered this in my most recent post to you. Given two ordinals, it's always the case that one is an element of the other or vice versa. — fishfry

OK, this makes more sense than what you told me in the other post, that one "precedes" the other. You are explaining that one is a part of the other, and the one that is the part is the lesser..

This is NOT true of sets in general, but it IS true for ordinals, and that's what makes the construction work. — fishfry

I assume that an ordinal is a type of set then. It consists of identifiable elements, or parts, some ordinals being subsets of others. My question now is, why would people refer to it as a "number"? Say for instance that "4" is used to signify an ordinal. What it signifies is a collection of elements, some lesser than others. By what principle is this group of elements united to be held as an object, a number? Do you know what I mean? A set has a definition, and it is by the defining terms that the sameness of the things in the set are classed together as "one", and this constitutes the unity of the set. In the case of the "ordinals", as a set, what defines the set, describing the sameness of the elements, allowing them to be classed together as a set?

No, as I'm pointing out to you. It's true that every ordinal is cardinally equivalent to itself, but that tells us nothing. You're trying to make a point based on obfuscating the distinction between cardinal numbers, on the one hand, and cardinal equivalence, on the other. — fishfry

The issue, which you are not acknowledging is that "cardinal" has a completely different meaning, with ontologically significant ramifications, in your use of "cardinally equivalent" and "cardinal number".

Let me explain with reference to your (I hope this is acceptable use of "your") hand/glove analogy. Let's take the hand and the glove as separate objects. Do you agree that there is an amount, or quantity, of fingers which each has, regardless of whether they have been counted? The claim that there is a quantity which each has, is attested by, or justified by, the fact that they are what you call "cardinally equivalent". So "cardinal" here, in the sense of "cardinally equivalent" refers to a quantity or amount which has not necessarily been determined. Suppose now, we determine the amount of fingers that the hand has, by applying a count. and we now have a "cardinal number" which represents the amount of fingers on each, the glove and the hand. In this sense "cardinal" refers to the amount, or quantity which has been determined by the process of counting.

Other way 'round. A cardinal number is defined as a particular ordinal, namely the least ordinal (in the sense of set membership) cardinally equivalent to a given set. — fishfry

Do you agree with this characterization then? An ordinal is a type of set, and a cardinal is a type of ordinal. Logical priority is given to "set". So do you agree that a cardinal number is not an object, but a collection of objects, as a set? Or, do you have a defining principle whereby the collection itself can be named as an object, allowing that these sets can be understood as objects, called numbers?

Right. I can live with that. I know I have the same number of fingers as my glove, but I don't know how many fingers that is. — fishfry

But this is an inaccurate representation. What you are saying, in the case of "cardinal numbers", is not "that. I know I have the same number of fingers as my glove, but I don't know how many fingers that is", but that there is no "number" which corresponds with the amount of fingers in my glove, until it has been counted and judged. You can say, I know I have the same "amount" of fingers as my glove, but you cannot use "number" here, because you are insisting that the number which represents how many fingers there are, is only create by the count.

Cardinal equivalence is a relation between two sets. It's not something a set can have by itself. — fishfry

But you already said a set can be cardinally equivalent with itself. "If nothing else, every ordinal is cardinally equivalent to itself, so the point is made."

I see where you're going with this. Given a set, it has a cardinal number, which -- after we know what this means -- is its "cardinality." You want to claim that the set's cardinality is an inherent property. But no, actually it's a defined attribute. First we define a class of objects called the cardinal numbers; then every set is cardinally equivalent to exactly one of them. But before we defined what cardinal numbers were, we couldn't say that a set has a cardinal number. I suppose this is a subtle point, one I'll have to think about. — fishfry

Yes this exemplifies the ontological problem I referred to. Let's say "cardinality" is a definable attribute. Can we say that there is a corresponding amount, or quantity, which the thing (set) has, regardless of whether its cardinality has been determined? What can we call this, the quantity of elements which a thing (set) has, regardless of whether that quantity has been judged as a number, if not its "cardinality"?

But when you got up that morning, before you came to my party, you weren't a room 3 person or whatever. The assignment is made after you show up, according to a scheme I made up. Your room-ness is not an inherent part of you. — fishfry

I see this as a very dangerously insecure, and uncertain approach, epistemically. See, your "scheme" is completely arbitrary. You may decide whatever property you please, as the principle for classification, and the "correctness" of your classification is a product simply of your judgement. In other words, however you group the people, is automatically the correct grouping.. The only reason why I am not a 3 person prior to going to the party is that your classification system has not been determined yet. If your system has been determined, then my position is already determined by my relationship to that system without the need for your judgement. It is your judgement which must be forced, by the principles of the system, to ensure a true classification. My correct positioning cannot be consequent on your judgement, because if you make a mistake and place me in the wrong room, according to your system, you need to be able to acknowledge this. and this is not the case if my positioning is solely dependent on your judgement.

If you go the other way, as you are doing, then the position is determined by your subjective judgement alone, not by the true relation between the system of principles and the object to be judged. So if you make a mistake, and put me in the wrong room, because your measurement was wrong, I have no means to argue against you, because it is your judgement which puts me in group 3, not the relation between your system and me.

The definition of 'precedes' ('less than') had been given by fishfry many posts ago. To have missed it is to have not paid attention to the posts.

Df. If x and y are ordinals, then x precedes y (x is less than y) iff x is an element of y.

Th. If y is an ordinal, then for all x, if x is a member of y, then x is a subset of y.

Th. If y is an ordinal, then y = {x | x is an ordinal & xey}. I.e., every ordinal is the set of its preceding ordinals.

/

Ordinals are called 'ordinal numbers'. But that is not needed for the actual formal results in set theory. We could just as easily always say 'ordinal' instead of 'ordinal number'. However, 'ordinal number' does reflect that there are operations on ordinals - ordinal arithmetic, ordinal multiplication, and ordinal exponentiation.

'is a number' is not a predicate of set theory. Saying 'ordinal number' rather than 'ordinal' does not confer any special property of being a "number"; it's just suggestive phrasing and does not add any inferential force.

/

There is no set of all the ordinals.

/

'is an ordinal' was defined, at least a few times, already in this thread.

/

fishfry is using 'cardinally equivalent' to mean 'equinumerous' (also 'equipollent'). the term 'cardinally equivalent' does not depend on having previously defined 'is a cardinal' nor 'the cardinality of'.

Again:

df. x and y are cardinally equivalent iff there is a bijection between x and y
[neither 'ordinal' nor 'cardinal' are mentioned in that definition]

Probably a more common terminology is:
df. x and y are equinumerous iff there is a bijection between x and y

df. x is a cardinal iff (x is an ordinal & ~Ey(yex & x and y are equinumerous))

df. card(y) = the cardinal x such that x and y are equinumerous

th. Ax(x is a natural number -> x is an ordinal)

th. {x | x is a natural number} = {x | x is an ordinal & x is finite

/

IMPORTANT SUGGESTION:

One should always understand that English terminologies such as 'ordinal number', 'equinumerous', etc. are only nicknames for actual formal symbols in set theory. The nicknames are suggestive of certain intuitions and motivations, but the nicknames do not confer any deductive effect. The only things that definitions that provide only abbreviation of actual formulas written solely in the primitive language.

It is a common error for people who are not familiar with formal theories to think that we can make deductions from the themes suggested in the nicknames; for example, saying "Well, if they're ordinal numbers then they must have all the properties of other numbers such as natural numbers, rational numbers, et. al}. That's wrong.

So, again, it is incorrect to think that 'cardinally equivalent' has some kind of "ontological" connection with 'is a cardinal' or 'the cardinality of'.

Again, three separate things (and the suggestiveness of the English word 'cardinal' in each of them does not confer any deductive force):

x and y are equinumerous [fishfry is calling that 'x and y are cardinally equivalent']

x is a cardinal

x is the cardinality of y [or said by, 'x = card(y)']

and:

th. x is a cardinal iff Ey x = card(y) [and 'x is a cardinality' just means 'x is a cardinal']

/

Every set is an object. Every cardinal is a set. Every cardinal is an object.

But 'object' is not a term in the language of set theory. It's just a word we use to talk about the things (objects) that are ranged over by the quantifier in the primitive language.

When talking about Z set theories, one can use 'set' and 'object' interchangeably since Z set theories don't "talk about" anything other than sets.

Yes you are correct, it's cleaner to not use proof by contradiction. — fishfry

Is that a thing? Ok.

But @TonesInDeepFreeze doesn't appear to be eschewing proof by contradiction, instead merely proving (still by contradiction) a stronger denial of surjectivity than mere failure of surjectivity. Hence his supposing ("toward contradiction") that the denial is false amounts to supposing less than necessarily complete surjectivity, which would entail the whole power set being in the range of f, and amounts instead to supposing merely the presence of S in the range of f.

Thus showing, that the naturals can't be used to keep count of their own groupings/combinations/sub-sets if any of them (naturals) are needed to index (I mean count, map to) groupings they aren't in. Because that would create the set S, which would need to but couldn't without contradiction be in the range of f.

Hope I've got that right. And if I have, then the suggestion to prove only the more specific failure of surjectivity has helped me, at least.

Then again, Tones hasn't exactly signposted the supposition, that S is in the range of f (the supposition being, I thought, in order to show that it leads to contradiction), and he doesn't even explicitly state it. It just (as line 4 perhaps alludes but doesn't actually say) follows from

EyeX f(y) = S — TonesInDeepFreeze

which (I guess?) follows from

Let f(y) = S — TonesInDeepFreeze

So, I don't know. Maybe they were undecided whether it was meant to be read as a proof by contradiction or not.

Is that a thing? — bongo fury

For some mathematicians its a stylistic preference.(I'm not sure, but I think maybe my version is Cantor's version.)*

proving (still by contradiction) a stronger denial of surjectivity than mere failure of surjectivity. — bongo fury

The proofs prove the exact same result - nothing more nothing less, except for a stylistic choice. Both proofs are correct and intutionistically valid.

EyeX f(y) = S
— TonesInDeepFreeze

which (I guess?) follows from

f(y) = S
— TonesInDeepFreeze — bongo fury

That's backwards.

EyeX f(y) = S
Let f(y) = S

That's existential instantiation.

EyeX f(y) = S is itself an RAA premise within the proof. In that way, firshfry's RAA is deferred in my proof to later.

I didn't write EyeX f(y) = S as a separate line, since I didn't belabor certain obvious steps; it's not a fully formal proof.

/

RAA and modus tollens are basically the same. RAA as a rule in natural deduction(or a derived rule from a Hilbert style system) while modus tollens does the job as an axiom in a Hilbert style system.

/

* Re Canor's diagonal proof, if I recall correctly, he doesn't start with a premise that there is a surjection and then derive a contradiction to infer there is not a surjection. Rather, he reasons about any arbitrary enumeration and shows that it is not a surjection. And I've read certain mathematicians say that they prefer not to set it up as an RAA. Same with the infinitude of the primes and other results. I don't know any philosophical reason for that; I take it as a stylistic choice.

Incorrect: We should not use 'least' if we don't mean quantity.

It is typical of cranks unfamiliar with mathematical practice to think that the special mathematical senses of words most conform to their own sense of the words or even to everyday non-mathematical senses. The formal theories don't even have natural language words in them. Rather, they are purely symbolic. Natural language words are used conversationally and in writing so that we can more easily communicate and see concepts in our mind's eye. The words themselves are often suggestive of our intuitions and our conceptual motivations, but proofs in the formal theory cannot appeal to what the words suggest or connote. And for any word such as 'least' if a crank simply could not stomach using that word in the mathematical sense, then, if we were fabulously indulgent of the crank, we could say, "Fine, we'll say 'schmleast' instead. 'schmardinality' instead'. 'ploompty ket' instead of 'empty set' ... It would not affect the mathematics, as the structural relations among the words would remain, and the formal symbolism too.
— TonesInDeepFreeze — Metaphysician Undercover

That incorrectly makes it appear that I said, "Incorrect: We should not use 'least' if we don't mean quantity."

The proofs prove the exact same result - nothing more nothing less, — TonesInDeepFreeze

Sure, but yours begins (read as a proof by contradiction) by denying a more specific claim of failure of surjectivity: the claim that such sets as, in particular, S will fail to be in the range of f. Was my point. Obviously that more specific claim of failure implies the more general. But the denial of it is weaker than the denial of the more general. I thought this might be the correct way to interpret

We don't need to suppose toward contradiction that there is a surjection. — TonesInDeepFreeze

(That you were saying we can suppose less.)

That's backwards. — TonesInDeepFreeze

Sure. I was prepared to guess at quantifier introduction on the backwards journey, but the "I guess" probably sounded sarcastic. Without the sarcasm it probably doesn't improve much.

In that way, fishfry's RAA is deferred in my proof to later. — TonesInDeepFreeze

Yes. It's still a proof by contradiction, just not so upfront.

I didn't write EyeX f(y) = S as a separate line, since I didn't belabor certain obvious steps; it's not a fully formal proof. — TonesInDeepFreeze

Sure. And it's still a proof by contradiction.

RAA and modus tollens are basically the same. — TonesInDeepFreeze

By RAA here I take it you mean the whole argument, while earlier it was a tag for the line that you

suppose toward contradiction — TonesInDeepFreeze

? Cool.

RAA and modus tollens are basically the same. — TonesInDeepFreeze

Another reason not to expect an important contrast in your reworking.

For some mathematicians its a stylistic preference. — TonesInDeepFreeze

signpost[ing] — bongo fury

yours begins (read as a proof by contradiction) by denying a more specific claim of failure of surjectivity: the claim that such sets as, in particular, S will fail to be in the range of f. — bongo fury

In the beginning, I didn't deny any claim claim whatsoever.

And, of course, I wouldn't even think of denying the claim that S is not in the range of f.

In that way, fishfry's RAA is deferred in my proof to later.
— TonesInDeepFreeze

Yes. It's still a proof by contradiction, just not so upfront. — bongo fury

Yes, that is fair to say. Notice, I didn't say that RAA would be avoided in a (natural deduction style) proof. I only mentioned a particular opening RAA premise, viz. "There is a surjection" and that we don't need to adopt it as an RAA premise.

And it's still a proof by contradiction. — bongo fury

It has an RAA nested within. Of course.

RAA and modus tollens are basically the same.
— TonesInDeepFreeze

Another reason not to expect an important contrast in your reworking. — bongo fury

As far as I can tell, the difference is merely stylistic. I said that in my previous post.

/

For reference:

Let f:X -> PX
Let S = {y e X | ~y e f(y)}
S e PX
S e ran(f) -> EyeX f(y) = S
Let f(y) = S
y e f(y) <-> ~ y e f(y)
So f is not a surjection

And, of course, I wouldn't even think of denying the claim that S is not in the range of f. — TonesInDeepFreeze

Except in a line properly signposted as RAA.

And, of course, I wouldn't even think of denying the claim that S is not in the range of f.
— TonesInDeepFreeze

Except in a line properly signposted as RAA. — bongo fury

There's a double negative in what you're saying. RAA premise would not need to deny ~P. Rather, in this case, the premise is P.

Anyway, that's not the beginning of my proof.

Yes you are correct, it's cleaner to not use proof by contradiction.
— fishfry

Is that a thing? Ok. — bongo fury

It's perhaps a little cleaner in terms of exposition. Not a big deal either way.

There's a double negative in what you're saying. — TonesInDeepFreeze

Of course.

RAA premise would not need to deny ~P. Rather, in this case, the premise is P. — TonesInDeepFreeze

Exactly, if for some reason you want to label the RAA line "P" rather "~P".

In a line properly signposted as RAA, and in a discussion in which someone had bothered to say

We don't need to suppose toward contradiction that there is a surjection. — TonesInDeepFreeze

it could make sense to display under that signpost (P or ~P depending on signposting preferences, or a form of words such as I chose so that the question didn't arise) the denial of what is to be shown in the argument. This denial will be the supposition toward a contradiction.

What is to be shown is that S can't, without contradiction, be in the range of f.

So the denial, the suitable RAA line, the supposition toward contradiction, if you or anyone did want to belabor the point, or understand the point about "not needing to suppose toward contradiction..." is indeed

"S is in the range of f",

and it might be interesting that this is taking the place of "f is surjective", in a proof by contradiction.

Anyway, that's not the beginning of my proof. — TonesInDeepFreeze

Agreed.

Maybe they were undecided whether it was meant to be read as a proof by contradiction or not. — bongo fury

Not a big deal — fishfry

How true.

I can see it both ways.

Starting with an RAA premise provides a clear structure.

Not starting with RAA, but instead talking about an arbitrary function from S to PS, is a little more arch, and reveals the hammer blow more by surprise.

I wonder why Cantor didn't start with RAA.

In any case, the choice is not to be confused with eschewing intutionistically invalid RAA, since it is intuitionistically valid.

RAA premise would not need to deny ~P. Rather, in this case, the premise is P.
— TonesInDeepFreeze

Exactly, if for some reason you want to label the RAA line "P" rather "~P". In a line properly signposted as RAA, and in a discussion in which someone had bothered to say

We don't need to suppose toward contradiction that there is a surjection.
— TonesInDeepFreeze — bongo fury

I don't know what you're saying.

it could make sense to display under that signpost (P or ~P depending on signposting preferences, or a form of words such as I chose so that the question didn't arise) the denial of what is to be shown. This denial will be the supposition toward a contradiction. What is to be shown is that S can't, without contradiction, be in the range of f. So the denial, the suitable RAA line, the supposition toward contradiction, if you or anyone did want to belabor the point, or understand the point about "not needing to suppose toward contradiction..." is indeed "S is in the range of f", and it might be interesting that this is taking the place of "f is surjective", in a proof by contradiction. — bongo fury

You lost me.

No, you can't be bothered, and why should you.

My bad. Carry on.

I did bother. I read your post three times but couldn't figure it out.

I don't know whether this bears on anything here, but just in case, there is huge difference between:

(1) ~~P RAA premise ... contradiction ... infer ~P

and

(2) P RAA premise ... contradiction ... infer ~P.

(1) is not intuitionistically valid. (2) is intuitionistically valid.

My mistake. My proof does not use RAA.

I see that what I left tacit is not RAA, but just modus tollens:

Let f:X -> PX
Let S = {y e X | ~y e f(y)}
S e PX
S e ran(f) -> EyeX f(y) = S
Suppose EyeX f(y) = S [for -> introduction]
Let f(y) = S
y e f(y) <-> ~ y e f(y)
EyeX f(y) = S -> (y e f(y) <-> ~ y e f(y)) [-> introduction]
~EyeX f(y) = S [intuitionistically valid modus tollens]
So f is not onto PS

Another instance in which modus tollens does the same job as RAA.

That incorrectly makes it appear that I said, "Incorrect: We should not use 'least' if we don't mean quantity." — TonesInDeepFreeze

That is what you said. You said the phrase, "We should not use 'least' if we don't mean quantity" is incorrect. I asked, if you don't mean some sort of quantity then what do you mean by "least". And fishfry gave me an answer to that.

OK, this makes more sense than what you told me in the other post, that one "precedes" the other. You are explaining that one is a part of the other, and the one that is the part is the lesser.. — Metaphysician Undercover

Ok good.

I assume that an ordinal is a type of set then. — Metaphysician Undercover

Yes. But that should be no surprise. In set theory everything is a set. There are no urelements in standard set theory. In math every single thing is a set. Numbers, groups, topological spaces, cardinals, ordinals, are all sets. Sets whose elements are sets whose elements are sets, drilling all the way down to the empty set. There is nothing but sets. Of course one need not found math on set theory, but in standard math, that's how it's done. Everything is a set.

It consists of identifiable elements, or parts, some ordinals being subsets of others. — Metaphysician Undercover

Yes. And those elements are sets. and those sets' elements are sets, all the way down to the empty set. Everything is a set. That's why they say math is based on set theory. Of course that's only historically contingent. Are numbers "really" sets? That's the question raised (and answered in the negative) by Paul Benacerraf in his famous essay, What Numbers Could Not Be.

My question now is, why would people refer to it as a "number"? — Metaphysician Undercover

There's no general definition of number. Negative numbers didn't use to be regarded as numbers, nor did zero, irrational numbers, complex numbers. Quaternions are numbers these days, but William Rowan Hamilton got famous for discovering/inventing them in 1843. When Cantor introduced his cardinals and ordinals he got a lot of pushback from the mathematical community of the day, but in the end his point of view won out, and the transfinite ordinals and cardinals are numbers. What is a "number" is a matter of historical contingency.

Say for instance that "4" is used to signify an ordinal. What it signifies is a collection of elements, some lesser than others. — Metaphysician Undercover

Ok. And lets be perfectly explicit. In the formalism, $0 = \emptyset, 1 = \{0\}, 2 = \{0, 1\}, 3 = \{0, 1, 2\}$ and $4 = \{0,1,2,3,\}$. Of course this is only a formalism. As Benacerraf points out, the number 4 can't really "be" this set. Rather, it's just a particular representation. The number 4 is the abstract thingie pointed to by the representation. But we've had this conversation before.

By what principle is this group of elements united to be held as an object, a number? — Metaphysician Undercover

Groups of elements are united to be held as sets by the axioms of set theory. If X is a set and Y is a set then their union and intersection are sets, and so forth. You can find the axioms here.

Now what is an "object," I don't know, because object is a term of art in computer programming but not in math. And what's a number is, as I've pointed out, a matter of historical contingency. There are no principles other than Planck's great observation that scientific progress proceeds one funeral at a time. Meaning that the old guard die off and the young Turks grow up taking the new ideas for granted.

Do you know what I mean? — Metaphysician Undercover

Yes. You want to know what entitles $\omega^\omega$ to number-hood. Well it's the same thing that entitled $\sqrt 2, i$, and -47 to numberhood. Historical contingency. Someone said "Hey this weird thing is useful, I'll call it a number," and everyone else said, "You're crazy," and a generation or two later everyone called it a number. Simple as that. Human opinion over time, nothing deeper than that.

A set has a definition, and it is by the defining terms that the sameness of the things in the set are classed together as "one", and this constitutes the unity of the set. — Metaphysician Undercover

No, not at all. First, the elements of a set need not be "the same" in any meaningful way. The only thing they have in common is that they're elements of a given set. The concept of "set" itself has no definition, as I've pointed out to you in the past. A particular set might have a specific definition; but even that sometimes fails, as in the nonconstructive sets given by various set-theoretic axioms. A set exists when the axioms say it does. To take a non-mathematical example, the set consisting of the number 5, the tuna sandwich I had for lunch, and the Mormon Tabernacle Choir may be taken together into a set consisting of three elements. Of course in math you can't have examples like that; the elements of sets have to be other sets. Unless you are working in a set theory that has urelements, which is a bit of a niche area.

In the case of the "ordinals", as a set, what defines the set, describing the sameness of the elements, allowing them to be classed together as a set? — Metaphysician Undercover

There is no set of ordinals, this is the famous Burali-Forti paradox.

What makes a particular set an ordinal is that it satisfies the textbook definition of an ordinal, namely a transitive set well-ordered by $\in$. That technical definition needs to be unpacked, but that's the definition. If a set satisfies that definition, it's an ordinal.

The issue, which you are not acknowledging is that "cardinal" has a completely different meaning, with ontologically significant ramifications, in your use of "cardinally equivalent" and "cardinal number". — Metaphysician Undercover

I've been pointing out to you the different meaning of cardinal equivalence and cardinal number for several posts now. I'm not sure why you claim I am not "acknowledging" that difference. I have been expending quite a few keystrokes to explain that distinction to you.

Let me explain with reference to your (I hope this is acceptable use of "your") hand/glove analogy. Let's take the hand and the glove as separate objects. Do you agree that there is an amount, or quantity, of fingers which each has, regardless of whether they have been counted? The claim that there is a quantity which each has, is attested by, or justified by, the fact that they are what you call "cardinally equivalent". So "cardinal" here, in the sense of "cardinally equivalent" refers to a quantity or amount which has not necessarily been determined. Suppose now, we determine the amount of fingers that the hand has, by applying a count. and we now have a "cardinal number" which represents the amount of fingers on each, the glove and the hand. In this sense "cardinal" refers to the amount, or quantity which has been determined by the process of counting. — Metaphysician Undercover

I get the point you're making, it's an interesting philosophical point. If I define an odd number as a number that leaves a remainder of 1 when integer-divided by 2; and I then prove that 47 is an odd number; was 47 an odd number before I made the definition? It's a good question in mathematical philosophy. Not one we'll solve here today.

Do you agree with this characterization then? An ordinal is a type of set, and a cardinal is a type of ordinal. — Metaphysician Undercover

Well not exactly. An ordinal is a type of set, yes. But a cardinal is not a "type" of ordinal at all. Rather, among all the ordinals cardinally equivalent to a given set, we take the least of them and designate that as the set's cardinal. So the cardinal-ness of an ordinal is not a property of an ordinal; rather, it's a name we give to an ordinal that has a particular property relative to a lot of other ordinals. Subtle point but important. It's a little like the captain of a football team. The captain is not a "type" of player; rather, the captain is a player that we have designated as the captain. The ordinal definition of cardinal is like that.

Logical priority is given to "set". — Metaphysician Undercover

In the sense that in the set-theoretic formalization of math, sets are fundamental. Ok. If that's what you're saying.

So do you agree that a cardinal number is not an object, but a collection of objects, as a set? — Metaphysician Undercover

I don't know what an object is (except in the context of everyday English, or computer programming; but not in math); so you'll have to tell me.

But a cardinal number is a set, yes. Everything is a set in set theory. Everything is a set.

Or, do you have a defining principle whereby the collection itself can be named as an object, allowing that these sets can be understood as objects, called numbers? — Metaphysician Undercover

I have no idea what you mean by object. I only know about sets. The defining principles of what can be called sets are the axioms of set theory. What's called a number is a matter of human agreement, often hard-won over generations and always subject to revision.

Even the same object in different contexts is or isn't a number. A classic example is the number $i$, the imaginary unit. We call that a number. But we can model the complex numbers as a particular set of 2x2 matrices, and then we call them matrices and not complex numbers. The question of what's a number is a matter of human convention. There is no general definition of number.

But this is an inaccurate representation. What you are saying, in the case of "cardinal numbers", is not "that. I know I have the same number of fingers as my glove, but I don't know how many fingers that is", but that there is no "number" which corresponds with the amount of fingers in my glove, until it has been counted and judged. — Metaphysician Undercover

Well of course that's an interesting philosophical question, which we will not solve today. Was 47 an odd number before I defined what an odd number is? You want to say that somehow the cardinal numbers existed before we defined them. Fine, you're a Platonist today. Sometimes I am too. Other times, not so much. What of it? I agree it's a good question. Whether the cardinal numbers were "out there" waiting for von Neumann to come along and give them their definition; or whether he made them up out of his productive mind.

After all, in other posts you have cast personal doubt on the very existence of mathematical sets; and now you want to claim that cardinal numbers were already out there waiting for von Neumann to come along. You see you're at best a part-time Platonist yourself.

You can say, I know I have the same "amount" of fingers as my glove, but you cannot use "number" here, because you are insisting that the number which represents how many fingers there are, is only create by the count. — Metaphysician Undercover

If I put on my Platonist hat, I'll admit that the number 5 existed even before there were humans, before the first fish crawled onto land, before the earth formed, before the universe exploded into existence, if in fact it ever did any such thing.

And then I'll ask, well if the number 5 existed before the universe did, where did it exist? What else might live there? The Baby Jesus? The Flying Spaghetti Monster? Captain Ahab? Platonism is hard to defend once you start thinking about it.

I must say, though, that I am surprised to find you suddenly advocating for mathematical Platonism, after so many posts in which you have denied the existence of mathematical objects. Have you changed your mind without realizing it?

Cardinal equivalence is a relation between two sets. It's not something a set can have by itself.
— fishfry

But you already said a set can be cardinally equivalent with itself. "If nothing else, every ordinal is cardinally equivalent to itself, so the point is made." — Metaphysician Undercover

Bit disingenuous there. The relation "has the same parents" is a binary relation, it inputs two people and outputs True or False. But they don't have to be distinct people. I have the same parents as myself.

Yes this exemplifies the ontological problem I referred to. — Metaphysician Undercover

Which I fully acknowledge, and note that we are not going to solve it here. Were the transfinite cardinals out there waiting to be discovered by Cantor and then formally defined by von Neumann?

But I must note that I find it very strange to see you suddenly advocating for mathematical Platonism, after denying the existence of mathematical sets.

Let's say "cardinality" is a definable attribute. Can we say that there is a corresponding amount, or quantity, which the thing (set) has, regardless of whether its cardinality has been determined? What can we call this, the quantity of elements which a thing (set) has, regardless of whether that quantity has been judged as a number, if not its "cardinality"? — Metaphysician Undercover

Heck of I know. Did the number 5 exist before the Big Bang? Was it out there waiting for humanoids with five fingers to come along? Maybe you can answer me that first, before you ask me about the transfinite cardinals.

I see this as a very dangerously insecure, and uncertain approach, epistemically. See, your "scheme" is completely arbitrary. You may decide whatever property you please, as the principle for classification, and the "correctness" of your classification is a product simply of your judgement. In other words, however you group the people, is automatically the correct grouping.. The only reason why I am not a 3 person prior to going to the party is that your classification system has not been determined yet. If your system has been determined, then my position is already determined by my relationship to that system without the need for your judgement. It is your judgement which must be forced, by the principles of the system, to ensure a true classification. My correct positioning cannot be consequent on your judgement, because if you make a mistake and place me in the wrong room, according to your system, you need to be able to acknowledge this. and this is not the case if my positioning is solely dependent on your judgement. — Metaphysician Undercover

I will agree that the fiveness of the fingers on my hand is not as arbitrary as my assignment of categories to people such as which room I put them in at a party, or who I designate as the captain of the team.

But @Meta, really, you are a mathematical Platonist? I had no idea.

If you go the other way, as you are doing, then the position is determined by your subjective judgement alone, not by the true relation between the system of principles and the object to be judged. So if you make a mistake, and put me in the wrong room, because your measurement was wrong, I have no means to argue against you, because it is your judgement which puts me in group 3, not the relation between your system and me. — Metaphysician Undercover

I agree with the points you're raising. I don't know if 5 existed before there were humans to invent math. I truly don't know if the transfinite cardinals were out there waiting to be discovered by Cantor, and formalized by von Neumann. After all, set theory is an exercise in formal logic. We write down axioms and prove things, but the axioms are not "true" in any meaningful sense. Perhaps we're back to the Frege-Hilbert controversy again.

My mistake.

inductive set well-ordered by ∈ — fishfry

I think you meant 'transitive set well ordered by ∈'.

I think you meant 'transitive set well ordered by ∈'. — TonesInDeepFreeze

Thanks, I made the correction.

Could someone rightfully say that 0, 1, and points are not in any sense sets? Or is there more too that?

https://plato.stanford.edu/entries/structuralism-mathematics/

Could someone rightfully say that 0, 1, and points are not in any sense sets? Or is there more too that? — Gregory

Sure. Euclid didn't have set theory but he talked about points. As far as the modern definition of numbers, there's Russell's type theory and its modern variants, there are category-theoret definitions, and so forth. I don't know much about any of these alternatives.

Benacerraf described it like this:

To be the number 3 is no more and no less than to be preceded by 2, 1, and possibly 0, and to be followed by 4,5, and so forth. And to be the number 4 is no more and no less than to be preceded by 3, 2, 1, and possibly 0, and to be followed by.... Any object can play the role of 3; that is, any object can be the third element in some progression. What is peculiar to 3 is that it defines that role -not by being a paradigm of any object which plays it, but by representing the relation that any third member of a progression bears to the rest of the progression.

That is, the number 3 is not an object at all. Rather, it's a thing defined by its relation to other numbers. In his famous essay he kicked off the field of mathematical structuralism. @TonesInDeepFreeze already gave this link, I'll repeat it for reference.

https://plato.stanford.edu/entries/structuralism-mathematics/

Benacerraf's essay can be downloaded here. The quote above is found on page 70,

https://documents.pub/document/benacerraf-what-numbers-could-not-be.html

You have to click Download then it makes you wait 60 seconds. Other online links to the article either make you read it online or else don't let you read it at all. When I'm in charge, academic paywalls will be abolished. Taxpayers already paid for this research. I looked it up. Benacerraf worked at Princeton and Princeton takes Federal money.

In category theory there's a thing called a natural numbers object which is intended to capture the structural essence of natural numbers. I don't know much about this and the Wiki article isn't particularly enlightening.

Here's an article about the natural numbers type in modern type theory. It's also not very enlightening unless one is a specialist.

So the bottom line is that structuralists don't think that natural numbers "are" sets; or even that natural numbers are any particular thing at all. A natural number is whatever relates to other things the way natural numbers do.

ps -- Here's the Wiki article on mathematical structuralism.

https://en.wikipedia.org/wiki/Structuralism_(philosophy_of_mathematics)

https://thephilosophyforum.com/discussion/comment/568807

With a Hilbert style system, the axiom we use to derive modus tollens is given in the intuitionistically invalid form:

(~P -> ~Q) -> (Q -.> P)

and then we derive the intuitionistically valid form:

(P -> Q) -> (~Q -> ~P)

The reason is that we can't derive the intuitionistically invalid form from the intuitionistically valid form.

I didn't know about structuralism in math! That the number one is an idea, a true idea, seems to me to be the basis of all that follows though, kinda that unity before the plurality. But structuralism in all forms is a really interesting idea!

I found the comments about Cohen's Filter in the article I linked fascinating. Like most math people I knew of his breakthrough results, but was unfamiliar with the actual math. I'd be interested to hear opinions from the set theorists on the forum about this. :cool: