## Taking from the infinite.

• 2.6k
I didn't get why you chose that clause in particular. I see now - it was just the nominated example.

Oh right, good point. Yes a "nominated example," great phrase. A synecdoche, as it were, a part that stands in for the whole, like "all hands on deck."

(Disclaimer yes, apology and placation no.)

I took it as very effective placation, so over-the-top I had to laugh.
• 1.8k
I'm looking at the notion of 'prior' syntactically (others may wish to discuss 'prior' in another sense, but then I'd like to know the definition of 'prior').

So a notation X (primitive or defined) is prior to another notation Y (defined) iff the definition of Y depends on X. (1) So this is relative to the sequence of definitions; different treatments of a theory, even with all the same set of defined notations, may have different sequences of definitions, (2) We need a definition of 'depends on'.

The notations that are defined for set theory are function symbols, predicate symbols, and variable binding operators such as the abstraction operator and the definite description operator. I'm leaving out the variable binding operators for now, because even giving a rigorous definition of the variable binding operators is complicated and requires double induction. So by 'notation' here I mean just function symbols and predicate symbols.

I haven't yet come up with a definition of 'depends on'. Intuitively it's that Y depends on all the notations that appear in the definiens for Y, and the notations that appear in the notations in the definens for Y, and finitely backwards until we reach the primitives. So it's inductive. And for a notation there's a tree, not a sequence, back to the primitives. For example in set theory:

n is even <-> (n is a natural number & Ek(k is a natural number & n = 2*k))

So 'even' depends on 'natural number', and '2' and '*'. And each of those depend on previously defined notations, and downwards in a tree to the primitives '=' and 'e'.

But 'even' (or any other notation) could also have been defined in the primitive language alone, without using any intermediary notations. This may make a non-syntactical notion of 'prior' problematic. For example, neither 'ordinal' nor 'cardinal' is non-syntactically prior since both could be defined themselves using only the primitive 'e'. Of course, in practice, the definition of 'cardinal' has 'ordinal' in the definiens. But that is not necessary, as 'cardinal' could also be defined from just '=' and 'e'. Of course such a definition of 'cardinal' would be a massive formula and impractical for people to work with. But 'practical' is not formalized, and what we are investigating is formal syntax. In principle, even if it would not be practical, 'cardinal' can be defined form 'e' alone.

Anyway, for 'prior' I need

Tree(Y) = [fill in formal definition of the tree of notations that branches up to the definition of Y]

X is prior to Y iff X is a node in Tree(Y)
• 1.8k
This is not circular:

df: K is a cardinal <-> (K is an ordinal & Aj(j e K -> there is no bijection between j and K))
["K is a cardinal iff (K is an ordinal and there is no bijection between K and an ordinal less than K"]

df: card(x) = the least ordinal j such that there is a bijection between x and j
["the cardinality of x is the least ordinal that has a bijection with x"]

theorem: Ax card(x) is a cardinal
["every cardinality is a cardinal"]

theorem: Aj(j is an ordinal -> EK card(j) = K)
["every ordinal has a cardinality"]

If we adopt a particular systematic and explicit sequence of definitions, and eschew locutions that don't "interock" with one another, then we leave fewer openings for being strawmanned by cranks.
• 1.8k
Regarding saying that two sets have the same cardinality without saying what that cardinality it is:

like saying that the score in a baseball game is tied -- without saying what the score is.

That's really good.
• 1.8k
Just to say proactively:

Some people claim that classifications must obey certain essentialities in order to be correct. For example (I'm not trying to state the more complicated actual zoological taxonomy) a claim that only this classification is correct:

animal (mammal (canine, feline, ...), reptile (snake, lizard ...) ,,,)

But that notion is not viable. For example:

(1) passenger vehicle (Ford (Fusion, Mustang, ...), Honda (Accord, Civic ...), ...)

(2) passenger vehicle (sedan (Ford, Honda, ...) van (Ford, Honda, ...), ,,,)

Both (1) and (2) may be pertinent depending on our purpose.

And we may be wary of the essentialist mistake in mathematics.
• 2.6k
Here is an interesting article about cardinality. Seems appropriate for the ongoing discussion.

Continuum Hypothesis?
• 1.8k

That article is good because it's hard to find layman's terms explanations of forcing and the proposed axioms.

But a couple of points:

"Cantor realized that [the set of natural numbers is 1-1 with the set of odd numbers]".

He "realized" it? It was known for at least 300 years. And probably a lot longer.

"In addition to the continuum hypothesis, most other questions about infinite sets turn out to be independent of ZFC as well."

Questions aren't independent. Sentences are. There are only countably many sentences. So the cardinality of the set of theorems is equal to the cardinality of the set of independent sentences.

/

"Kennedy, for one, thinks we may soon return to that “prelapsarian world.” “Hilbert, when he gave his speech, said human dignity depends upon us being able to decide things in mathematics in a yes-or-no fashion,”"

I hadn't read the speech. That is really interesting about human dignity. I understand why we would seek a decision procedure, but why would Hilbert think our dignity depends on it? So incompleteness would lead Hilbert to abandon hope of human dignity? This is really interesting.
• 4.4k
"Cantor realized that [the set of natural numbers is 1-1 with the set of odd numbers]".

But there are infinite more natural numbers, just as with the reals. Is the point that there are far more infinities of reals than infinity of naturals vs the odd? I can imagine putting any two infinities one to one if you start with one number, then two, and onward
• 1.8k
But there are infinite more natural numbers, just as with the reals.

Please. If you have a rigorous definition of "infinite more" different from set theoretic "greater cardinality" then fine, state your definition, and your claim could be right relative to that definition. Meanwhile the card(N) = card(set of odds) is a theorem of set theory.

Is the point that there are far more infinities of reals than infinity of naturals vs the odd?

I can't parse that.

I can imagine putting any two infinities one to one

It is a theorem of set theory that no set is 1-1 with its power set. It's a theory of set theory that N is not 1-1 with R.

If you have an alternative theory, then state your axioms. What you merely imagine is not mathematics.
• 4.4k
If you have an alternative theory, then state your axioms.

It seems to me the natural numbers are a type of power set to the odd. If I imagine (not a bad word) any infinity as a ruler going off east into forever, I can make this ruler and a second by taking the first number of a countable set and the first number of a uncountable set and send them off to infinity like you do with naturals and odds.Why isn't it the same thing? The rulers would fit side by side because they are all within infinity
• 1.8k

Impressionistic descriptions are fine for stoking creativity in mathematics and sometimes for making certain mathematical concepts intuitive. But they are not mathematical demonstrations.
• 4.4k
Impressionistic descriptions are fine for stoking creativity in mathematics and sometimes for making certain mathematical concepts intuitive. But they are not mathematical demonstrations.

If you know why we can do:

Odd numbers: 1 3 5 6....

Natural numbers: 1 2 3 4...

but can't do:

Countable: 1 2 3 4...

Uncountable: 1 2 3 4...

then say it.

I really want to know
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By definition, there is no bijection between a countable set and an uncountable set.

By theorem, there is no bijection between N and R. The proof has been given thousands of times in textbooks, articles, and on the Internet (it's even outlined in the article linked to above). You really are not familiar with the proof?

I really want to know

Then open an Internet search engine and type 'proof uncountability reals'.
• 4.4k
You really are not familiar with the proof?

No because I've asked people many times and they bring up the diagonal thing, although this just shows there are infinity more uncountable than countable and yes, however there are infinity many natural than odd. But you can biject with one and not the other? I'm not a jerk, just want some way I can understand what they are saying. It seems to me infinity is always just infinity at the end
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You are terribly confused. You asked me to prove there is not bijection between a countable set and an uncountable set. And I told you where to find the proof. But you say you already know about it. So there was no point in asking me.

But you say there are infinitely many more odd numbers than natural numbers. So YOU prove that. Not mind pictures, but mathematical proof.

If you really do want to understand, then get a book on set theory and start reading it from page 1. I could type proofs and explanations for you all day, but if you don't have the background for it, then it's a waste. Any proof I give you will depend on proofs and definitions previous to that proof, on and on backwards until we reach the axioms. So that is utterly impractical in a thread. The reasonable and enlightened way is to get a book and read it from page 1, from the axioms through the proofs.
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It seems to me you can state the reason you can move the odd numbers in line with the naturals but can't move the countable in line with the uncountable in a pretty simple way say that I can go and find more about this
• 2.6k
I've asked people many times and they bring up the diagonal thing, although this just shows there are infinity more uncountable than countable and yes, however there are infinity many natural than odd. But you can biject with one and not the other? I'm not a jerk, just want some way I can understand what they are saying. It seems to me infinity is always just infinity at the end

Have you seen the simple and beautiful proof of Cantor's theorem? It shows that there is no possible surjection from a set to its powerset.

Here's the proof. Let $X$ be a set, $\mathscr P(X)$ its powerset, the set of all subsets of $X$. Let $f: X \to \mathscr P(X)$ be a function, and by way of starting a proof by contradiction, suppose $f$ is a surjection.

Since $f$ is a function that inputs an element of $X$ and outputs some subset of $X$, for any given element $x \in X$ it may or may not be the case that $x \in f(x)$. Let $S$ be the subset of $X$ defined by $S = \{x \in X : x \notin f(x)\}$.

Now $S$ is a subset of $X$; and since by assumption $f$ is a surjection, there must be some element $s \in X$ such that $f(s) = S$.

Now we ask the question: Is $s \in S$? Well if it is, by definition it isn't; and if it isn't, by definition it is. Therefore the assumption that there is such an $s$; that is, that $f$ is a surjection; leads to a contradiction.

Therefore there is no surjection from any set to its powerset.

To take an example, consider the set {a,b,c}. Its powerset is {ø, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}}. It's perfectly clear that there's no surjection from a 3-element set to an 8-element set; and this principle holds even in the infinite case.

Cantor's theorem immediately gives us an endless hierarchy of infinities $\mathbb N, \mathscr P(\mathbb N), \mathscr P( \mathscr P(\mathbb N)), \dots$ In terms of the simplicity of the argument versus the profundity of the result, I don't think there's anything comparable in all of mathematics.
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Wow that's simply put. Thanks
• 2.6k
Wow that's simply put. Thanks

You are welcome! So glad that worked for you. This argument is much simpler and more natural than the diagonal argument, it should be better known.
• 1.8k

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

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
• 10.6k
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.

OK Tones, explain to me then what "least" means in "the mathematical sense", if it is not a quantitative term. It can't be "purely symbolic" in the context we are discussing. For example, when fishfry stated von Neumann's definition of a cardinal as "the least ordinal having that cardinality", through what criteria would you determine "least", if not through reference to quantity?

df: K is a cardinal iff K is an ordinal and there is no ordinal j less than K such that there is a bijection between K and j.

There is no mention of 'cardinal' or 'cardinality' in the definiens.

Here's another example. Look at your use of "less than". How is one ordinal "less than" another, without reference to quantity?

You wouldn't call it "my" theory of relativity, or "my" theory of evolution, just because I happened to invoke those well-established scientific ideas in a conversation.

Yes, in the context of the example we are discussing, I would. Unless you were quoting it word for word from another author, or explicitly attributing it to someone else, I would refer to it as your theory. I believe that is to be expected. Far too often, Einstein's theory, and Darwin's theory are misrepresented,. So instead of claiming that you are offering me 'Cantor's theory', it's much better that you acknowledge that you are offering me your own interpretation of 'Cantor's theory', which may have come through numerous secondary sources, unless you are providing me with quotes and references to the actual work.

It's a bit like saying that the score in a baseball game is tied -- without saying what the score is. Maybe that helps.

OK, so let's start with this then. In general we cannot determine that a game is tied without knowing the score. However, if we have some way of determining that the runs are equal, without counting them, and comparing, we might do that. Suppose one team scores first, then the other, and the scoring alternates back and forth, we'd know that every time the second team scores, the score would be tied, without counting any runs. Agree? Is this acceptable to you, as a representation of what you're saying?

If one thing is defined in terms of some other thing, the latter is logically prior. As is the case with cardinal numbers, which are defined as particular ordinal numbers.

Here's where the problem is. You already said that there is a cardinality which inheres within ordinals. This means that cardinality is a property of all ordinals, it is an essential, and therefore defining feature of ordinals. So we have a sense of "cardinality" which is logically prior to ordinals, as inherent to all ordinals, and we also have a sense of "cardinal" number which is specific to a particular type of ordinal.

I'd agree that given some ordinal number, it's cardinally equivalent to some other sets. It doesn't "have a cardinality" yet because we haven't defined that. We've only established that a given ordinal is cardinally equivalent to some other set.

Don't you see how this is becoming nonsensical? What you are saying is that it has a cardinality, because it is cardinally equivalent to other sets, but since we haven't determined its cardinality, it doesn't have a cardinal number. In essence, you are saying that it both has a cardinality, because it is cardinally equivalent, and it doesn't have a cardinality because it's cardinality hasn't been determined, or assigned a number.

Let's look at the baseball analogy. We know that the score is tied, through the equivalence, so we know that there is a score to the game. We cannot say that because we haven't determined the score there is no score. Likewise, for any object, we cannot say that it has no weight, or no length, or none of any other measurement, just because no one has measured it. What sense does it make to say that it has no cardinal number just because we haven't determined it?

Note per your earlier objection that by "least" I mean the ∈∈ relation, which well-orders any collection of ordinals. If you prefer "precedes everything else" instead of "least," just read it that way.

Actually, this explains nothing to me. "Precedes" is a relative term. So you need to qualify it, in relation to something. "Precedes" in what manner?

No. Cardinal equivalence is logically prior to ordinals in the sense that every ordinal is cardinally equivalent to some other sets. At the very least, every ordinal is cardinally equivalent to itself.

When you use the word "cardinality" you are halfway between cardinal numbers and cardinal equivalence, so you confuse the issue. Better to say that cardinal equivalence is logically prior to ordinals; and that (in the modern formulation) ordinals are logically prior to cardinals.

Yes, this demonstrates very well the problem I described above. Because the set has a "cardinal equivalence, it also necessarily has a cardinality, and a corresponding mathematical object which you call a cardinal number. Why do you think that you need to determine that object, the cardinal number, before that object exists as the object which it is assumed to be, the cardinal number?
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With my original question I didn't think too hard on the point that if one did take a drop of water from an infinite ocean they would have no place to take it. And if I created infinite land next to an infinite ocean that might create even more questions.

I was largely thinking in terms of the movie, "Dr. Strange" in which the Ancient One keeps her youth by stealing energy from the dark dimension which I was led to believe is an infinite world as perhaps our universe maybe as well. So basically 2 or more infinite universes at least in the fiction exists. The Ancient One had an ethical dilemma over her theft from the dark dimension even though in theory nothing is lost, but I wasn't sure. Many physicists believe there could be many worlds split into more worlds whenever a paradox needs to be resolved so Schroder's Cat can actually be both alive and dead, but in different worlds.
• 2.6k
We don't need to suppose toward contradiction that there is a surjection.

Yes you are correct, it's cleaner to not use proof by contradiction. Thanks for the clarification.
• 1.8k
k = the-least-ordinal_x such that Px <-> (k is an ordinal & Pk & Ah(h e k -> ~Ph))
• 2.6k
With my original question I didn't think too hard on the point that if one did take a drop of water from an infinite ocean they would have no place to take it. And if I created infinite land next to an infinite ocean that might create even more questions.

In general, a set can be infinite yet not include "everything." For example there are infinitely many even numbers, but they don't include the odd numbers.

In the case of an infinite ocean we have to work a little harder to get a good visualization. Maybe the world is like a 3D chessboard, with an infinite ocean on one level and an infinite plane of land on the next. So we can be sitting on land and reach down to take a cup of water from the infinite ocean below. Now we have a cup of water; but since the ocean is liquid, it immediately fills up the space where we removed the water, and there's still an infinite ocean.

What do you think? Visualization-wise, I mean?

The essential idea though is that you can always take a finite amount from an infinite set, and the set is still infinite. But it doesn't necessarily have to be all of what it was before. I believe @TheMadFool gave this example earlier, where we can start with the infinite set 1, 2, 3, 4, ..., then remove 1 to leave 2, 3, 4, ... What's left is still infinite, yet it's missing 1. That can happen too. Infinity is funny that way.

ps Here's another idea. The world is a flat, infinite plane. Like "flat earth" theory except instead of a great wall of ice around the edge, it just goes on forever. The entire world is a vast ocean, but there are infinitely many finite-sized islands spread throughout. So there's an island here and another one there, infinitely many in all, but they're all separated by water. So if you're on land you can always dip your cup in the infinite ocean.
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I guess we might need to add time to infinity if there is a process by which an infinite ocean refills itself? Or perhaps virtual particles that are trigger happy to become real? Lol. And I assume any place within an infinite ocean must be the center? Therefore any ripples will never reach the shore because there will never be one and therefore no entropy is added to this world? Or can ripples or motion itself exist in this ocean?
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Or can ripples or motion itself exist in this ocean?

If you drop a pebble in the ocean it will ripple forever. And if you drop lots of pebbles there will be lots of ripples, all the time.
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https://thephilosophyforum.com/discussion/comment/568451

We would continue to prove that the uncountability of Pw implies the uncountability of R:

It suffices to prove that the interval [0 1] is uncountable.

We have the theorem Ax Px 1-1 with 2^x.

So P^w 1-1 with 2^w.

And we prove 2^w injects in [0 1]
• 13.9k
I believe TheMadFool gave this example earlier, where we can start with the infinite set 1, 2, 3, 4, ..., then remove 1 to leave 2, 3, 4, ... What's left is still infinite, yet it's missing 1. That can happen too. Infinity is funny that way.

For finite sets,

1. K = {a, b, c}, L = {a}

2. Set difference: K - L = {b, c}

3. K - L =/= K

Where n(A) is the number of elements in set A,

n(K) = 3

n(L) = 1

4. Arithmetic difference: n(K) - n(L) = 3 - 1 = 2

5. n(K) - n(L) =/= n(K)

---------------------

For infinite sets,

6. N = {1, 2, 3,...} O = {1}

7. Set Difference: N - O = {2, 3, 4,...}

8. N - O =/= N

Where n(A) is the number of elements in set A,

n(N) = Infinity

n(O) = 1

9. Arithmetic difference: n(N) - n(O) = Infinity - 1 = Infinity

10. n(N) - n(O) = n(N)
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OK Tones, explain to me then what "least" means in "the mathematical sense", if it is not a quantitative term. It can't be "purely symbolic" in the context we are discussing. For example, when fishfry stated von Neumann's definition of a cardinal as "the least ordinal having that cardinality", through what criteria would you determine "least", if not through reference to quantity?

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. So for ordinals $\alpha$ and $\beta$, we define $\alpha \lt \beta$ if it happens to be the case that $\alpha \in \beta$. This is perfectly well-defined and unambiguous, especially in the case of the von Neumann ordinals which are constructed exactly so that this works out. $\emptyset \in \{\emptyset \} \in \{{\emptyset, \{\emptyset\}\}} \in \dots$, and that's exactly how ordinal "less than" is defined. It's also the case that the von Neumann ordinals are defined in such a way that $\in$ is transitive; if $\alpha \in \beta$ and $\beta \in \gamma$ then $\alpha \in \gamma$. This is NOT true of sets in general, but it IS true for ordinals, and that's what makes the construction work.

Here's another example. Look at your use of "less than". How is one ordinal "less than" another, without reference to quantity?

Via the $\in$, which is a primitive in set theory and is always true between any two ordinals in one direction or the other, by construction of the ordinals.

Yes, in the context of the example we are discussing, I would. Unless you were quoting it word for word from another author, or explicitly attributing it to someone else, I would refer to it as your theory. I believe that is to be expected. Far too often, Einstein's theory, and Darwin's theory are misrepresented,. So instead of claiming that you are offering me 'Cantor's theory', it's much better that you acknowledge that you are offering me your own interpretation of 'Cantor's theory', which may have come through numerous secondary sources, unless you are providing me with quotes and references to the actual work.

The Wikipedia articlea on ordinals, cardinals, Cantor, etc. are perfectly satisfactory in this regard.

OK, so let's start with this then. In general we cannot determine that a game is tied without knowing the score. However, if we have some way of determining that the runs are equal, without counting them, and comparing, we might do that. Suppose one team scores first, then the other, and the scoring alternates back and forth, we'd know that every time the second team scores, the score would be tied, without counting any runs. Agree? Is this acceptable to you, as a representation of what you're saying?

Rather than try to save that example, I'll just repeat the hand/glove example. By putting on a glove, I can determine whether my hand-fingers are in bijective correspondence with the glove-fingers, without knowing the actual cardinal number.

c
Here's where the problem is. You already said that there is a cardinality which inheres within ordinals. [/quote]

I went to great pains to note last time that your use of the word "cardinality" is ambiguous and causing you to be confused. You should either say cardinal equivalence or cardinal number, to clearly disambiguate these two distinct but related notions. Every ordinal number is inherently cardinally equivalent to many sets. If nothing else, every ordinal is cardinally equivalent to itself, so the point is made.

But we still don't know which cardinal number that is.

And again, when you say "cardinality," you obfuscate the distinction between these two concepts.

This means that cardinality is a property of all ordinals, [/qmote]

No no no no no. No. Every ordinal is cardinally equivalent
to many other sets, including itself. But when we clarify this terminology, your sophistic point evaporates.

it is an essential, and therefore defining feature of ordinals.

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.

So we have a sense of "cardinality" which is logically prior to ordinals, as inherent to all ordinals,

No no no no no. I hope I've explained this.

and we also have a sense of "cardinal" number which is specific to a particular type of ordinal.

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.

I hope you can see that by carefully using the phrases, "cardinal equivalence" and "cardinal number" properly, all confusion goes away. You are deliberately introducing confusion by using the word "cardinality" ambiguously.

Don't you see how this is becoming nonsensical?

No, it's very carefully thought out by 150 years worth of mathematicians including von Neumann, widely agreed to be "the smartest man in the world" by his contemporaries. It's your insistence that everyone else is wrong about things that you aren't willing to put in the work to understand that's nonsensical.

What you are saying is that it has a cardinality,

No. I am repeatedly telling you to stop using that word, because you are using it to confuse yourself. It's true that every ordinal is cardinally equivalent to various other sets including itself; and it's true that a cardinal number is defined as a particular ordinal.

Your entire argument is based on obfuscating the word cardinality. You should stop, because I can only explain this to you so many times without losing patience. I already explained this to you repeatedly in my previous post.

because it is cardinally equivalent to other sets, but since we haven't determined its cardinality, it doesn't have a cardinal number.

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.

In essence, you are saying that it both has a cardinality, because it is cardinally equivalent,

No. I'm telling you to stop using the word cardinality until you understand what's being talked about.

and it doesn't have a cardinality because it's cardinality hasn't been determined, or assigned a number.

This is you just continuing to confuse yourself over the word cardinality. If you'll just carefully say cardinal equivalence when you mean that, and cardinal number when you mean that, we might make progress.

Let's look at the baseball analogy. We know that the score is tied, through the equivalence, so we know that there is a score to the game. We cannot say that because we haven't determined the score there is no score. Likewise, for any object, we cannot say that it has no weight, or no length, or none of any other measurement, just because no one has measured it. What sense does it make to say that it has no cardinal number just because we haven't determined it?

Because cardinal numbers are a defined term[/url]. Given a set, we have to build a sophisticated technical apparatus in order to define what we mean by its cardinal number.

But I could take a step back from all this. My remark about what's logically prior to what is true, but it's not that important in the scheme of things. It's more important for you to make an effort to understand what ordinal numbers are, because they're important. So if all you care about it to be right about the logically prior business, that's the wrong thing to care about. It's not an important matter.

Actually, this explains nothing to me. "Precedes" is a relative term. So you need to qualify it, in relation to something. "Precedes" in what manner?

Given two ordinal numbers, it's always the case that one is an element of the other, as sets. We define $\lt$ as $\in$. If you prefer you can always think of the $\in$ whenever I say that one ordinal is "less than" another, or that some ordinal is the "least" with such and so property.

Yes, this demonstrates very well the problem I described above. Because the set has a "cardinal equivalence,

Cardinal equivalence is a relation between two sets. It's not something a set can have by itself. So it is not true that "a set has a cardinal equivalence" in isolation. That makes no sense. I can say, "My brother and I have the same parents," but it makes no sense to say, "I have the same parent." As what?? Having the same parent is a relation between two things. It doesn't stand alone. Likewise two sets may be cardinally equivalent to each other or not. But a set doesn't have cardinal equivalence by itself, that makes no sense.

it also necessarily has a cardinality,

Please stop using that word till you have a better grasp on the material. Your constant misuse of it is only causing you confusion.

and a corresponding mathematical object which you call a cardinal number.

Yes, it has that, after we've built up a whole bunch of theory to define what that is.

Why do you think that you need to determine that object, the cardinal number, before that object exists as the object which it is assumed to be, the cardinal number?

Because after Cantor defined cardinal equivalence, the question came up among philosophers, "What actually is a cardinal?" At first they did the obvious thing, they said a cardinal was the entire class of all the sets that are cardinally equivalent to a given set. That is a perfectly satisfactory definition, but it suffers from the flaw that such a class is not a set. Von Neumann figured out how to define cardinal numbers as particular sets, so that they could be manipulated using the rules of set theory.

It's just a matter of wanting to define what a cardinal number is, in formal terms. So that when we have two sets that are cardinally equivalent to each other, we have a collection of canonical sets such that every set is cardinally equivalent to exactly one of them.

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.

Here's an example. Whenever I have a party I like to put everyone in separate rooms according to their approximate height. I have ten rooms and I arrange the people so that there's more or less an equal number of people in each room. So at the party, each person is a "room 1 person" or a "room 2 person" and so forth.

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.

Likewise, given a set we can assign it a cardinal number. But it's far from clear that this is an inherent property of a set. Rather, we set up the scheme of defining cardinal numbers so that given a set, we can figure out which cardinal it's assigned to. It's an after-the-fact defined assignment, not an inherent quality.

Another example, a bunch of people show up for a work detail. I assign some to dig ditches, some to cut down trees, some to supervise, some to do this, some to do that.

Before I made the work assignments, the jobs were not inherent properties of the people. Rather, I assigned those jobs after the fact. Just as I can take a set and assign it a cardinal number. But of course in the case of a set's cardinal number, that's a more subtle question. Did the set "inherently" have a cardinal number before I assigned it?

Good point, if that is your point. I'll give this some thought.
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