Statistics

# A Note On Mathematical Axioms (With Regard To Sizes Of Infinity)

This isn’t easy going, but there’s a story out, with a very good write up in Quanta, on mathematical axioms, as they apply to different sizes of infinity. Key paragraph:

Their proof [about sizes of infinities], which appeared in May in the Annals of Mathematics, unites two rival axioms that have been posited as competing foundations for infinite mathematics. Aspero and Schindler showed that one of these axioms implies the other, raising the likelihood that both axioms — and all they intimate about infinity — are true.

For the moment, call these two supposed axioms D and E. The idea is that, before, people believed D and E separately, and now, it is thought that starting with D we can deduce E.

Assume that that’s true. Then, starting at D, and bringing in a whole host of other axioms, and necessary truths deduced from those axioms, such as are commonly found in logic and mathematical logic, E can be shown to follow from D—a shorthand way of saying it. More completely: E can be shown to follow from D and A_1, A_2, A_3, …, A_p, other associated axioms, and B_1, B_2, B_3,…B_q, necessary truths deduced from those A_i.

I don’t know the size of the entire suite of As and Bs is, but it will be substantial.

Regular readers will know an axiom, a proper axiom, is a proposition that is believed but cannot be proved. That is, it cannot be deduced from other axioms, which include truths of logic and so forth. True axioms are believed based on intuition alone.

Here’s a for-instance from the article, where you have to know “(*)” (the whole thing with parentheses) is one of the axioms mentioned above:

“With two highly productive axioms floating around, proponents of forcing faced a disturbing surplus. ‘Both the forcing axiom [Martin’s maximum] and the (*) axiom are beautiful and feel right and natural,’ Schindler said, so “which one do you choose?'”

There is nothing wrong this Schindler’s sentence; indeed, sentiment is how we judge all axioms. On how they feel, where that word is used in its intuitionist sense. All of our most important truths cannot be proved, but must be believed—on faith, if you like.

If the axioms contradicted each other, then adopting one would mean sacrificing the other’s nice consequences, and the judgment call might feel arbitrary. “You would have had to come up with some reasons why one of them is true and the other one is false — or maybe both should be false,” Schindler said.

Godel’s theorem lurks under this, but for us it’s not a mystery.

The curious thing for us is that sentence at the beginning, ” Aspero and Schindler showed that one of these axioms implies the other, raising the likelihood that both axioms — and all they intimate about infinity — are true.”

“Raising the likelihood” means adding probative information to the right hand size of a probability “equation”. Not all probabilities are numbers, so “equation” is used metaphorically.

Incidentally, all this has to do with continuum hypothesis. This is that there is no size of infinity between the infinity of the countables (1,2,3…) and the reals (e, π, 0.000000000001212,…). (I’m not the only one to dislike the names of these things.)

It is known the size of the reals is larger than the size of the countables. This is also put by saying the cardinality of the reals is bigger. And it was proposed that no other kinds of infinities could fit between the countables and reals. The new supposition, flowing from being able to prove one axiom from another, is that at least one other kind of infinity can be sandwiched in.

Funny thing is probability as a formal branch of math, i.e. measure theory, only uses countables and reals. There are known to be infinities larger than reals (and none smaller than countables), so

Categories: Statistics

### 54 replies »

1. Simon Platt says:

So…

Don’t leave us in the lurch! Things were just warming up.

2. Pk says:

If we have two axioms and one implies the other, then the other has been shown to not be an axiom. That is, since it can now be proved from another axiom. The probability that the second is true is 1 given that the first is true. ????

3. Sheri says:

Go ahead, leave us in the lurch. I’m OK with that.

4. Pk says:

Feels like we’ve been in the lurch for 18 months. Adding another, this one an infinite lurch, (second order infinite do loop) might be fun.

5. Jan Van Betsuni says:

Agreeing with Pk (comment above) ~ that: If both are AXIOMS are true ~ then they are not different ~ but at best just tangents on some common “observation/experience” derived through intuition. If an “AXIOM” merely reciprocates another statement (another “AXIOM”) with the same conclusion this is not additive proof. I sniff an academic scuffle over intellectual property rights to infinity. A brewing teapot tempest.

6. Pk says:

I should have added that if the first axiom was false, it could still lead to the second or it could not. Either way would not tell us if the second axiom was true or false.

7. Joy says:

Funny thing is probability as a formal branch of math, i.e. measure theory, only uses countables and reals. There are known to be infinities larger than reals (and none smaller than countables), so

Isn’t that a statement of the obvious? Or some kind of logical tautology?
If they are countable, let alone smaller than what can be counted, they’re not infinite!

8. Hagfish Bagpipe says:

”The cardinality of the reals is bigger.”

“Bigger” — that’s racist.

9. Ye Olde Statistician says:

I recollect a problem from one of my old topology texts I read back in the ’70s, as follows:
2. Neither the Axiom of Choice (AC) nor the Continuum Hypothesis (CH) can be proven from ZF a la Goedel’s Theorem.
3. Not only that, but if you take AC as axiomatic, CH cannot be proven from ZF+AC.
4. But if you take CH as axiomatic, AC can be proven from ZF+CH.

B’gawd, Whether AC is an axiom or not depends on what model you start with. Now, where have we heard something like that before? P(E|M)?

10. Shecky R says:

well, at least it’s assumed there are no smaller infinities than the countables…

11. Heresolong says:

“If they are countable, let alone smaller than what can be counted, they’re not infinite!”

Nope. An example of countable infinity is all the whole numbers. They go on forever but you can start working your way through 1, 2, 3, … You just don’t have time to finish.

Non-countable infinity, on the other hand, is just a set you can’t count through. The example given is real numbers. You can’t count them because in between each two real numbers is an infinite number of other real numbers. Example: Between 1 and 2 is 1.5, but between 1 and 1.5 is 1.25, but between 1 and 1.25 is 1.125, etc.

On the other question, I love countable and real. I like real because the other half is imaginary. Means that the quantity of all numbers is twice as big as the quantity of real numbers. I love to mess with my high school students when we are talking about infinity and then I point out that there are larger values of infinity. Many of them actually get a real kick out of it when they start to visualize the explanation.

12. Joy says:

“Nope. An example of countable infinity is all the whole numbers. They go on forever but you can start working your way through 1, 2, 3, … You just don’t have time to finish.”

Nope, if you can’t count’em they aren’t countable without using the imagination again.

because you don’t have the time!

so it’s the same situation for this:
“Non-countable infinity, on the other hand, is just a set you can’t count through. The example given is real numbers. You can’t count them because in between each two real numbers is an infinite number of other real numbers. Example: Between 1 and 2 is 1.5, but between 1 and 1.5 is 1.25, but between 1 and 1.25 is 1.125, etc.”
“In between” doesn’t make any difference.

What you’re saying is about a definition, which is not the same as the logic of being able to count.
If you can count real numbers but then say you don’t have enough time it’s the same situation with the in -betweens! If you’re saying
“that’s the definition”
Then that’s another matter. It still doesn’t make it sensible. That’s the nature of axioms isn’t it?

13. Joy says:

Counting the in-betweeners is the same problem as counting the whole numbers to infinity

However,
the Phrase,
“none smaller than countables”
is still a statement of the obvious.
If they’re smaller than that which is countable, they’re not infinities, so, there are none!

14. Joy says:

If they’re smaller than what can be counted, they’re not infinite!”

Number ten, number three!
both smaller than what can be counted
Infinity is just the limit of the imagination when intuition tells you that there’s no end to whatever the proposition at hand.

15. Heresolong says:

Joy, you are arguing with a definition. I simplified it to make it easier but since it didn’t seem to help here’s the actual definition.

A countably infinite set is one in which each element of the set can be mapped one to one onto the set of natural numbers.

That makes much less sense to the non-mathematician than what I said but it works out to the same thing. You map the set of whole numbers by assigning 1 to 1, 2 to 2, etc. Therefore whole numbers are a countably infinite set.

You can’t map real numbers to the set of natural numbers because, as I pointed out, as soon as you map 1 and 2, there is something in between. Every time you map two particular real numbers, there are more in between that haven’t been mapped. Therefore your statement that “it is the same with the in-betweens” isn’t accurate to the definition.

So I still take issue with your original comment “if they are countable, then they aren’t infinite”. That is not the definition of countable infinity.

16. Heresolong says:

“Infinity is just the limit of the imagination”

So are you saying that Calculus is only needed by people who have insufficient imagination? ?

17. Rudolph Harrier says:

I am enjoying this new side of Joy. Same ranting via free association as always, but in the realm of mathematics it’s even more blatantly bankrupt than usual.

18. Leo says:

Is the idea to found all of mathematics on set theory?

I remember seeing functions being defined as a certain kind of subset of the cross product. If f:A?B then f was identified as the subset F of A×B, the set of ordered pairs (a, b) with a?A and b?B, with the property that if (a, b_1) ? F and (a, b_2) ? F then b_1 = b_2.

And then the ordered pair was defined as the set, if I remember correctly, {{a}, {a, b}}.

So using set theory as a foundation covers at least functions.

Take a closer look at a ? A. Just as plus sign in 2+3 can be thought of as a function of 2 arguments +(2, 3) yielding a number, the membership predicate can be thought of as a function, again of 2 arguments, yielding a value of true or false. Membership is fundamental to sets, so defining ? in terms of ordered pairs is circular, and won’t do.

So in addition to a primitive and irreducible notion of set, do we also need something like that for functions?

19. Leo says:

Forgive the repost, but the symbols got garbled above.

Is the idea to found all of mathematics on set theory?

I remember seeing functions being defined as a certain kind of subset of cross product. If f:A arrow B then f was identified as the subset F of A x B, the set of ordered pairs (a, b) with a in A and b in B, with the property that if (a, b_1) in F and (a, b_2) in F then b_1 = b_2.

And then the ordered pair was defined as the set, if I remember correctly, {{a}, {a, b}}.

So using set theory as a foundation covers at least functions.

Take a closer look at a in A. Just as plus sign in 2+3 can be thought of as a function of 2 arguments +(2, 3) membership predicate can be thought of as a function, again of 2 arguments, yielding a value of true or false. Membership is fundamental to sets, so defining in(a, A) in terms of ordered pairs is circular, and won’t do.

So in addition to a primitive and irreducible notion of set, do we also need something like that for functions?

20. Joy says:

Heresolong,
Thanks for the responses,

Yes I am arguing with a definition! this is a place, where that kind of thing happened, or always used to. Maths used to be fun!

My attitude is more playful than it seems some are able to take.

The imagination and infinity? It doesn’t follow that just because I think the imagination or the human intellect cannot ‘see’ infinity, that it means anything beyond that. It is simply another way of saying what you said about ‘mapping’ or ‘counting’. Infinity can’t be pinned down

Although I understand what you said in your original and second post, where you clarify, I still *being honest, sorry, think what I said initially is correct. Not because I think I know better than you, of course, I just can’t see that the definition is without fault. It seems it requires some faith

21. Joy says:

Roodie,
You’re as rude and trollish as always

“Some ranting via free association as always”
That remark is nonsense but no doubt you had a ‘free’ thrill, from writing it
Go and bother someone else.

22. Joy says:

“So are you saying that Calculus is only needed by people who have insufficient imagination? ?”

Seems that you took this to be a slight, rather than a comment on the human mind in general.

23. philemon says:

Joy, dear, I know you mentioned that you lived in the UK somewhere, and had something to do with the NHS. But I’m curious, I can’t help it, where exactly are you located?

24. philemon says:

I don’t want to pry, of course, but, after all, one likes to know?

25. Phil says:

He probably fell asleep in the middle of the last sentence

26. KenS says:

Heresolong
The “there are always an infinite number of numbers between any two others” by way of explanation or to get a sense of why the real line is uncountable – does not work. Consider the line of rationals, which are countable (according to definitions).

27. Briggs says:

Holy moly you’re right.

My enemies are now deleting entire passages!

28. McChuck says:

@Ken – The rationals aren’t countable, because each numerator is supplied with an (integer-order) infinite number of denominators (except zero, which, although easily writable, is an asymptote). It’s a size of infinity between integers and reals.

@Heresolong – The imaginary numbers expand the number real number line into the complex plane, making an (real-order) infinity of complex numbers for each corresponding real number. (This is probably the secret to neutrino oscillation and electron spin.)

29. Joy says:

Heresolong,

“A countably infinite set is one in which each element of the set can be mapped one to one onto the set of natural numbers.”
(For “mapped” one might also use the word count or ‘clock’ or “collect”!!)

1
“You map the set of whole numbers by assigning 1 to 1, 2 to 2, etc. Therefore whole numbers are a countably infinite set.”
2
You can’t map real numbers to the set of natural numbers … as soon as you map 1 and 2, there is something in between. Every time you map two particular real numbers, there are more in between that haven’t been mapped.“
Therefore your statement that “it is the same with the in-betweens” isn’t accurate to the definition.”

Maybe it isn’t identical but the cause is the same, just more immediately obvious with the second / ‘real’/ in betweener example. One has to accept the definition of ‘count’ and then agree on where it breaks down. It just does it sooner with the ‘real’ numbers.

“So I sill take issue with your original comment
I don’t blame you!

“if they are countable, then they aren’t infinite”. That is not the definition of countable infinity.”
The problem is that counting is something that happens in your head, if you’re not a Mathematician. When you force it to happen on a ‘map’ or a physical thing not imagined, it all looks different again!

30. Ye Olde Statistician says:

Consider the line of rationals, which are countable (according to definitions).

Right, because the actual definition is the ability to place (e.g.) the rationals in 1:1 correspondence with the natural [counting] numbers; that is, countable not counted.
https://simple.wikipedia.org/wiki/Countable_set
Countable infinity is symbolized as ‘aleph-sub=null’ which I will abbreviate with the Latin letter as A_0. It turns out that
A_0+A_0=A_0 and
A_0*A_0=A_0, but
A_0^A_0 (exponentiation) >A_0. This was designated A_1.
The infinity of the real number line was discovered to be greater than A_0. It was designated c (for continuum). The Continuum Hypothesis is that c=A_1, and it has been famously proven to be unprovable, at least in ZF and ZF+AC.

D is dense in X is every open interval around an x contains a d. The rationals are dense in the reals. That is, every real number, like pi, can be approximated arbitrarily closely by a rational number (e.g., 22/7, 3.1, 3.14, 3.142, 3,1416, etc.) The irrationals are likewise dense in the reals, and polynomial functions are dense in the space of all complex-valued functions, which means that any function can be approximated arbitrarily closely by a polynomial function.

In my story “Places Where the Roads Don’t Go,” one of the characters proves that provable propositions are dense in the space of all logical propositions; that is, any unprovable theorem can be approximated arbitrarily closely by a provable one. This encourages a friend of his to take an awful, irreversible step.

The rationals aren’t countable, because each numerator is supplied with an (integer-order) infinite number of denominators

A countable infinity divided by a countable infinity is countably infinite; to wit:
1. 1/1
2. 2/1
3. 1/2
4. 1/3
5. 2/3
6. 4/1
7. 3/2
8. 2/3
9. 1/4
etc.
https://simple.wikipedia.org/wiki/Countable_set

31. Heresolong says:

Joy, I didn’t take it as a slight, nor did I take it personally. It’s math and it is fun. ?

However, you can’t dismiss a mathematical definition by saying “but it doesn’t apply if you aren’t a mathematician”. Countable infinity is a definition in mathematics. It makes no sense to even discuss it outside of the field of mathematics. So by saying “if it’s countable then it isn’t infinite” would be like saying “sure that’s a cow if you are a farmer, but for non-farmers it isn’t”. That’s not how definitions work.

As a final follow-up you can probably now see why I tried (twice) to simplify the definition of countable infinity. Because the real mathematicians (I’m an engineer who now teaches high school math up to and including Calculus) are now jumping in to “explain” aleph null sets and density of propositions. ?

However, I do like Ye Olde Statistician’s observation that countable does not mean counted. I will be filing that one away for the future.

PS I think Philemon asked where you lived because if you are in the UK then you use the metric system, which means that all the natural numbers have to be multiplied by 0.62 before you map the sets. ?

32. Joy says:

Heresolong,
How odd that you think there’s no point discussing infinities outside of mathematics!
If you’re teaching high school maths you need a sense of humour.

As for [okay with uncounted yet]? That is simply saying the same thing, about time and for how long the definition holds before a new one is required.

I did say,

If you’re saying?“that’s the definition”?Then that’s another matter. It still doesn’t make it sensible. That’s the nature of axioms isn’t it?

??The ensuing discussion proved the point

Philemon,
Once, a few years back, here, I gave an honest response about the dullest of topics… axioms… and received the dirtiest and most insinuating response from a then regular reader. He never came back, maybe other reasons, but it has to be said that telling the most innocent truth can often render the most bizarre reactions on the internet. I therefore take your question with the same kind of suspicion as is evidently the wiser option. If you are sincere, more information is elsewhere on the site, since I’ve been more candid than most.

33. DAV says:

Joy,

Hah! Uncountable does mean infinitely large or unimaginably large in everyday English. Your critics lack imagination and likely have trouble with English.

34. Milton Hathaway says:

YOS – Excellent comment. Thanks for introducing me to the concept of “dense in x”, that’s a very interesting way of thinking about such things, especially the extension into areas outside of math. I have a challenge that’s been puzzling me for a long time now, I’m going to try that approach, to approximate arbitrarily closely by related things that I do understand.

BTW, I’m not understanding “A_0^A_0 (exponentiation) >A_0”. It seems to me that the distinction between A_0 and A_1 is that the latter includes the set of irrational numbers. How do you get an irrational number by exponentiation of rational numbers? Hmmm, maybe I just answered my own question, since 2 ^ (1/2) is an irrational number. Never mind!

(Can you get pi or e by exponentiation of rational numbers? Can you get any irrational number by exponentiation of rational numbers? If not, does this imply that there is an infinite set between A_0 and A_1?)

Heresolong – you mention you are an engineer who now teaches math. As a fellow engineer, your back-and-forth with Joy brings to mind some conversations I’ve had with coworkers, especially non-engineers. I’d have a discussion with someone who very convincingly proposes a different approach to something I’m working on. I get back to my desk and start pondering the alternate approach that sounded so reasonable, and I soon realize I can’t figure out how it is actionable. Thinking that I’m just not understanding the proposal, I talk to the proposer again for clarification, and the process repeats. There are all sorts of possible reasons that I can’t do anything with the proposal, including that I’m just not smart enough, but I am paid to produce stuff, so I have to put it aside and move on (and if the proposer is your boss, some Dilbertian wisdom is required). In your case, now that you are teaching, I don’t know how one would handle such a situation when dealing with a persistent student.

35. Joy says:

Ah, Dav 🙂

Your command of English seems very English
Since looked up the thesaurus and one of the common English synonyms offered was

36. Ye Olde Statistician says:

I’m not understanding “A_0^A_0 (exponentiation) >A_0”.

It is that the usual arithmetic functions (+) and (*) do not affect the cardinality of a set. That it, sticking with countably infinite sets, if you add say the even integers to the odd integers the sum is still A_0. There are no more integers than there are odd integers only. If you multiply two countably infinite sets, the product is still countably infinite: A table with infinite columns and infinite rows has no more cells than it has columns. Since subtraction and division are simply addition and multiplication by an inverse, the same holds true if you take away a countable infinity from a countably infinite set, the remainder is countably infinite. Ditto if you divide: if you divide an integer by another integer, the ratio (i.e. rational number) is still countably infinite.

Only the fifth operation, exponentiation, raises the cardinality. Countable infinity raised to a countably infinite power is something greater than countable infinity, and is denoted A_1. Whether A_1 is the cardinality of the irrationals is an unprovable hypothesis. This does not mean that pi is the countably infinite power of some integer or anything like that. The aleph numbers are not numbers at all in the ordinary sense.

Here’s another notion: countable infinity is digital; uncountable infinity is analog.

37. Eric Quinnell says:

When considering infinity, rules need not apply — in mathematics, quantum physics, inside black holes, etc. Axioms may most certainly infer axioms in infinity land, as such hypothesis are neither testable nor provable! And I disagree, nothing is “known” about infinity, and certainly not more of this infinity than that.

As far as I’m concerned, “here be red dragons” and “here be green dragons” can most certainly co exist.

My question is which is more: the number of real infinities or the number of definitions of infinities that provide neither insight nor utility beyond the mere knowledge of…well…the infinite?

The best answer will win a grant or tenure. Or both!

38. Ken Seton at Greenhood - aka KenS says:

One must remember that whether or not a collection of reals is countable or uncountable (i.e. whether it can be placed into a one-to-one correspondence with the natural numbers) has nothing – that’s NOTHING – to do with the density of those numbers on the real line. A super dense set can be countable and a super sparse set can be uncountable. This is all about the definition. Which tells us that the designation of a collection as uncountable is all about – merely all about – whether or not there exists an algorithm capable of putting such a collection into the correspondence. We should not ever mislead students with any talk of the probability of picking a number or with allusions to the density of number.

Eric – If there is objectively only one infinity then clearly the number of definitions is the greater.

39. Joy says:

“However, you can’t dismiss a mathematical definition by saying “but it doesn’t apply if you aren’t a mathematician””
Of course you can.
There are hidden premises and assumptions in the arguments made above.
I like what Eric Q said. Thee definitely be dragons if you discuss infinity.

Re YOS’s digital or analogue,
Even that makes little difference to the problem.
Whichever method, it’s still chopping it into segments and you don’t even know what it is!
Fractions give a better illusion of accuracy but they’re always imagined and not really accurate.
Half a pice of cake isn’t half! Even when laser cut.

Fractions are better though, never serve 0.5 of a piece of cake, you might end up with five pieces in our house, if that’s a fairy cake, (cup cake in American) it’s not enough.

40. Heresolong says:

“How odd that you think there’s no point discussing infinities outside of mathematics!”

That isn’t even remotely what I said or implied, so if you are going to make arguments on strawmen that you have created then a productive and interesting discussion has just ended.

“If you’re teaching high school maths you need a sense of humour”

Since you don’t know me at all I don’t know where you got this from, so why you would engage in what appears to be an ad hominem attack is beyond me.

Dav: ” in everyday English”

Context is everything. This article was not about everyday English, it was about mathematics. So when Joy jumped in to explain that the definition was wrong because “everyday English”, she was completely missing the point. Instead she now appears to be arguing for the sake of arguing while changing the topic to make it harder to keep up with what she is apparently arguing for (or against).

Peace, out.

41. Joy says:

Heresolong,
No, I don’t know you and nor was I implying that [you] lacked a sense of humour.
However, that you took it personally seems apparent and was not remotely my intention.
If I said,
“One needs a sense of humour”
Does that make it any better or clearer?

““Maths is supposed to be fun”?”
I think it should be at high school age. For what it’s worth, I think Engineers make good teachers.

In no way did I imply a personal attack but see Rudolph’s comment and you might understand the ‘context’ of the dynamic, if you haven’t been here too long, you wouldn’t understand.

I too detected some impatience in your comment but that’s quite okay

Don’t back out of the discussion on my account I’ll do that for you and promise not to say anything else on this post, even if tempted,
Sorry and Solong, Heresolong

42. Yancey Ward says:

On the countable definition- here is the way I think about: can one count all the integers between 1 and 1 quintillion given enough time? The answer is obviously yes. Now: can one count all the real numbers between 1 and 2 given enough time. The answer is obviously no.

43. Rudolph Harrier says:

Yancey,

That doesn’t work, since it would be impossible to “count” all the rational numbers between 0 and 1 in any finite amount of time, yet the rational numbers between 0 and 1 form a countably infinite set. Or for that matter the natural numbers, i.e. the counting numbers, cannot be completely counted in any finite amount of time.

“Countable set” would probably be less misleading if it was called something like “Eventually countable set.” That is, we can’t actually count all the positive integers in a finite amount of time. But we can describe a process that will eventually count up to any number n, no matter how large n is, in a finite amount of time. In contrast for the real numbers no matter what process we choose some numbers will be excluded no matter how long we continue. But “countable” is easier to state quickly, and mathematicians are used to some terms not being as intuitive as they might first appear (ex. only some lines in the plane being “linear” subspaces, with “affine” being the word used to describe lines and similar sets in general.)

44. Rudolph Harrier says:

I think a lot of confusion arises from the fact that we intuitively think of counting as an ordering. “Successor functions” might be a bit heavy no the jargon side, but the idea that we find that the successor of 1 is 2 does line up with how we think about counting. This leads us to think about countable sets as a type of ordered discrete set, where there is always a clear “next” element with nothing in the “gaps” between numbers. We also think about the ordering as being from lower to higher. That is, 1 must come before 2, 2 must come before 3, etc.

Of course, in modern mathematics that way of looking at things doesn’t work at all, since the rational numbers are dense in the reals and do not have a countable ordering that goes from lowest to highest rational. That is, there is no “first smallest positive integer” or anything like that. Rather “countable” is more like “there is a way to order them after rearrangement” which already stretches our idea of “counting” pretty far. It’s only natural to think that way if you are analyzing things in terms of sequences, which historically was a given but in modern terms is not how most people first encounter the term “countable.”

45. Ye Olde Statistician says:

That doesn’t work, since it would be impossible to “count” all the rational numbers between 0 and 1

Unclear, I see, on the concept of “countable.” The only requirement is that the set can be placed in 1:1 correspondence with the natural numbers, and this is clearly true of the rationals. [Links have been previously provided.] No one ever said boo about completing the count. Infinity, remember? And infinity is not a “number.” You can never reach it even if you have an infinite amount of time in which to do it. And as Aristotle noted long before Cantor or Hilbert came along, a physically realized infinity is impossible.
Technical terms, whether of mathematics, theology, philosophy, or anything else, cannot be properly understood in the common tongue. Consider that in math a maze is a “simple” curve and a figure-8 is not, which goes against the vulgar notion of “simple.”

46. Ye Olde Statistician says:

there is no “first smallest positive integer”

1.
Anything smaller is not an integer.
There is no smallest positive real number, which is what I think you meant to write.

47. Paul Murphy says:

A fascinating discussion.

Several weeks ago I did the Barnaby Ridge hike in Alberta’s Castle Mountain park. One bit of this features a series of switchbacks that I lost count of for sweating and puffing –thus giving me a new understanding of both the infinite and the non denumerable.

48. Yancey says:

Rudolf,

I wrote real numbers, not rational numbers. I am distinguishing between countable and uncountable sets explicitly and trying to give an intuitive way of thinking about it. The real numbers include the integers, rational numbers, irrational numbers, the non-algebraic numbers, and the transcendtal numbers. In short, I can take any range and apply the “can one count all the rational numbers between (m through n)/(p through q) regardless of the gaps between n and m, and p and q- the answer is obviously yes, given enough time. That, to me, is the very essence of countable- within any range, they can be counted if you have the time to do so. However, once you get out of the rational numbers into the irrational and the transcedental numbers, that is no longer the case because you can’t define a range where the numbers within it are finite- you reach the infinity of the continuum at that point.

49. Rudolph Harrier says:

YOS,

In response to the first I did mean that it isn’t possible to count the rationales between 0 and 1 in a finite amount of time. But here I am not using the mathematical sense of “count” as “find a bijection to the natural numbers” but rather then more intuitive sense of counting as listing off things in order (i.e. “this is the first number, this is the second number,” etc.) That’s the sort of counting that Yancey is talking about so it made sense to keep to that definition in the response. After all, mathematical definitions aren’t proper for all situations and it’s not like there’s a universal sense of what any word in mathematics means anyway (I’ve already alluded to the distinction in “linear” between geometry and algebra, but “normal” is probably the worst.)

As for the second point that was a typo. I meant “first smallest positive rational.” That is, intuitively we expected counting to go from a number to the next greatest number, i.e. how 3 comes after 2 because the next largest integer after 2 is 3. The rationals don’t have this sort of ordering which is part of why it is hard to intuitively think of them as “countable” though of course from a modern set-theoretic point of view they are countable.

50. Ye Olde Statistician says:

mathematical definitions aren’t proper for all situations

Although they are when discussing mathematical concepts. You are the first person I’ve encountered who wants to define Cantor’s transfinities in terms of actual physical infinities.

And no, I’ve never thought of ‘counting’ and ‘ordering’ as being the same activity. I do not expect #2 widget to be the “next” widget in your sense. When counting the rationals, we generally go from 1 to 1/2 to 2/1 and so on.

And what’s wrong with ‘normal’? Are you complaining that reading mathematical texts requires old-fashioned ‘close reading’ rather than ‘browsing’ or ‘scanning’?

No wonder so many people don’t get the Cosmological Arguments.

51. DAV says:

Heresolong,

Context is everything

Indeed. Sometimes context shifts. I suspect you have a lot of trouble with puns.

52. Fr. John Rickert, FSSP says:

You’ve entered into a region, known as, “The Twilight Zone.” Insert cigarette-swinging Rod Serling here.

But seriously, as a mathematician, most people do not realize how intrinsically and inextricably mysterious mathematics is.

53. Allen Stoner says:

My argument is that all infinites must be equal, as there are only a certain number of sub atomic particles in the universe from which to create a representation of all units in said infinites. With that being the case, then each infinite can only be represented with what is available in the universe.

54. Eric Quinnell says:

“ Eric – If there is objectively only one infinity then clearly the number of definitions is the greater. “

Haha, indeed!

And what are “definitions” for that matter other than our mortal attempt to describe precisely that which is “by definition” unbounded?

Which is greater, the infinite of the cosmos and space-time or the concept of the infinite we hold in the space of our heads? The answer is “yes”.

We know here be dragons. Color them any definition you wish, you will not be incorrect.