Your post helped me to clarify my view.
If 'there are as many even integers as integers' is replaced with 'my new set has as many elements as E', then it makes sense.

Tim;

'indoctrinate'
cause to believe something: to teach somebody a belief, doctrine, or ideology thoroughly and systematically, especially with the goal of discouraging independent thought or the acceptance of other opinions
Somehow this word seems appropriate, if applied to yourself.
First, there are no experts, just some people with more experience than others.

The example in question:
N={1 2 3 4 5 6 7 8 ...}
E={2 4 6 8 10 12 14 16...}
A one to one correspondence of E to N, with the conclusion 'there are as many even integers as there are integers'.
Why are the same integers on both sides of the correspondence?
1 is linked to 2, and 2 is linked to 4, i.e. all even integers have 2 links, all odd have 1 link, and each integer is unique.

And with that you might show us how you can 'approach infinity'.

A one to one correspondence of E to N, with the conclusion 'there are as many even integers as there are integers'. — sandman

Informally, and for so long as you and your readers understand the formality underlying the informality, yes. More accurately, the cardinality of the two sets is the same.

Under a correct understanding of a subject, it is possible to be right as well as wrong. Under a wrong understanding, anything goes. The question, then, goes not just to conclusions drawn, but to the understandings they're based on.

Informally, and for so long as you and your readers understand the formality underlying the informality, yes. — tim wood

Not in direct opposition to this statement, but just a remark...

Some educators are stuck on introducing maximum controversy when they present their students with novel abstractions. Expressions such as "as many as", I think, have to be reserved to already familiarized audiences, but some teachers find it particularly amusing to thrust the "inside jokes" of mathematics directly to the uninitiated.

However, let's do something different. We take the same sets N and E. We know that N has the even numbers. So we pair the members of E with the even numbers in N. We can do that perfectly and with each member of E in bijection with the even number members of N. What now of the odd numbers in N? They have no matching counterpart in E.

Doesn't this mean N > E? — TheMadFool

Not it doesn't. Because it doesn't meet the definition of ">", here is the definition:

X > Y if and only if there is an injection from Y to X, but there is no injection from X to Y.

↪fishfry
Do you agree that there exists at least one bijection from E to N?
— fishfry

So there was nothing wrong when Socrates defined humans as featherless bipeds and someone came along with a chicken plucked of all its feathers and declared "this is a human"? After all there was/is at least one human that fit the definition. — TheMadFool

I can only implore you to carefully re-read what I and others have written. Perhaps Cantor's beautiful ideas will come to you at some point. Perhaps not.

I can only implore you to carefully re-read what I and others have written. Perhaps Cantor's beautiful ideas will come to you at some point. Perhaps not. — fishfry

:clap: :ok: :up:

I really hope so too.Thanks.

I would just flip bits along the diagonal and have a sequence they didn't include on their list. — ee

[Direction is not a factor in forming a sequence.]

One can think of it as a game with symbols. — Eee — ee

[A well known mathematician, whose name escapes me, when asked to define mathematics replied "a manipulation of symbols". I was impressed by such a concise and accurate description.]

a quote by Cantor, Source:  Ewald, W., From Kant to Hilbert, Oxford 1996.
"[… Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established.  In particular, in the introduction of new numbers, it is only obligated to give definitions of them which will bestow such a determinacy and, in certain circumstances, such a relationship to the other numbers that they can in any given instance be precisely distinguished.  As soon as a number satisfies all these conditions, it can and must be regarded in mathematics as existent and real
"… the essence of mathematics lies entirely in its freedom".]"
We know what Cantor thought.
Cantor was most concerned with his standing in the mathematical community, beyond his bouts of depression. In publishing papers on the nature of infinity through his transfinite numbers, he claimed it was a supernatural revelation. Maybe expecting acceptance through authority, which does occur even in modern times. The well known mathematicians of that period didn't accept his ideas initially, so skepticism is not new.
He associated the truly infinite with GOD. He should have left it there.

[Direction is not a factor in forming a sequence.] — sandman

The informal talk of 'flipping the bits along the diagonal' is just handy way of describing a function not on the list, $f : N \to \{0,1\}$. You have yet to show that you know enough about math to lecture me about sequences.

[A well known mathematician, whose name escapes me, when asked to define mathematics replied "a manipulation of symbols". I was impressed by such a concise and accurate description.] — sandman

Formalism is one way to think of things. It has its problems, but so do perhaps all -isms that try to tell the final truth. Math is only interesting, it seems to me, because of its practical and/or intuitive appeal. Even if we think of it from a distance as a system of dead symbols, we then have to reason intuitively about that system. In the same way a chess player has an intuitive grasp of what is going on, even as, importantly, a computer can check the legality of moves. This doesn't mean that kings and knights exist in some magical realm. But it's safe to talk about chess players sharing intuitions that make playing the game possible. This is just sufficiently intersubjective concepts, not unlike us talking now in a shared language.

Here's a great book:

https://www.amazon.com/Analysis-History-Undergraduate-Texts-Mathematics/dp/0387770313

Practice and intuition came before a relatively settled foundation. The real numbers are at the center of both practice and intuition. While you have focused on Cantor, the nature of the real numbers as theoretical entities is just as strange and philosophically questionable. Why aren't you railing against the 'superstition' that 2 has a square root? Have you ever seen it? It's a theoretical construction.

And Cantor's motivations or philosophy or theology, whatever one thinks of it, doesn't stick to his ideas. Unless you are also ready to call Newtonian physics a scam too,

https://en.wikipedia.org/wiki/Isaac_Newton%27s_occult_studies

Because these issues fascinate you, it's possible that you'll end up a mathematician. I was disturbed by Cantor once too, but I kept on thinking and reading and eventually got some formal education in the discipline. I suggest a book like this one: https://www.amazon.com/Concise-Introduction-Mathematics-Third-Chapman/dp/1439835985

I really hope so too.Thanks. — TheMadFool

Thank you for being so gracious. Not always my experience around here and I appreciate it.

Let me see if I can respond directly to your plucked chicken remark. In that case there is an underlying reality, namely the existence, or the fact, of human beings. Our existence on earth logically and historically precedes any formal description of the matter. So when we say, "A human is a featherless biped," we are NOT actually defining what it means to be human. A human is so much more complex than any mere formalization could encapsulate. So at best when we say "A human is a featherless biped," what we are doing is laying down the rules for a formal, logical treatment of what it means to be human. And if we are wise we will recognize that. Personally I believe in science but I oppose scientism.

Now when it comes to sets, the matter is totally different.

In this case there is no underlying reality. There is no "setness" that we are trying to formalize. Rather, sets are whatever satisfies the rules we're writing down. Before the rules are written down, there are no sets! Unlike with humans, whose existence precedes any humanly contingent theory of them.

So you may be thinking, "This bijection business might be logically correct, but it doesn't match my beliefs about sets." But the right way to think about it is: "The knight in chess moves the way it does because that is the rule. And sets behave the way they do because those are the rules!" We must learn to adjust our intuitions to the symbology. We have to cast off our intuitions and just do exactly what the symbols say. That's the essence of learning math.

[Now if you want to talk about the "unreasonable effectiveness" of math in the real world then that's another subject. But math must be taken on its own terms! It need not conform to reality nor does it, most of the time].

In high school they tell you that sets are collections, but as we all know Russell ruined Frege's day be discovering that this cannot possibly be. Ever since then, set is an undefined term just as point and line are undefined terms in Euclidean geometry.

If you want to know what a set is, the answer is that nobody has any idea what a set is. A set is whatever satisfies the rules of set theory. In fact set theorists like to mess around with different rules to see what kinds of crazy sets they can come up with.

So when we see a plucked chicken we may fairly say, "Oh, our formal characterization was good as far as it went, but it needs to be refined. How about featherless bipedal mammals?" Then someone will shave a gorilla and we'll be off to the races again. A formal system can never completely capture every aspect of reality.

Whereas with sets, that thought process doesn't happen because sets don't exist outside of the rules we stipulate for them. There is no underlying reality to appeal to. Sets are what the rules say they are. So when we say two sets are cardinally equivalent if there's at least one single solitary lonely bijection between them; then that is the absolute truth of the matter. Because in set theory, the rules precede the reality. With humans, the reality precedes the rules.

tl;dr: It's chess not chickens.

He associated the truly infinite with GOD. He should have left it there. — sandman

It's not such a bad association, although I do not know what you mean by it. I am under the impression that Cantor's major contribution wasn't inventing or discovering a concept called infinity - that was old news. What he did do was develop ideas of transfinite cardinals and ordinals along with an arithmetic of transfinite cardinal and ordinal numbers. That is, of different sized infinities, of which there are apparently a whole lot.

In this case there is no underlying reality. There is no "setness" that we are trying to formalize. Rather, sets are whatever satisfies the rules we're writing down. Before the rules are written down, there are no sets! — fishfry

Or as Charles Sanders Peirce aptly put it, mathematics is the science that draws necessary conclusions about purely hypothetical states of things.

What he [Cantor] did do was develop ideas of transfinite cardinals and ordinals along with an arithmetic of transfinite cardinal and ordinal numbers. That is, of different sized infinities, of which there are apparently a whole lot. — tim wood

Ironically, there is a countably infinite number of cardinals, only the smallest of which is itself countable.

Ironically, there is a countably infinite number of cardinals, only the smallest of which is itself countable. — aletheist

Is that correct? I thought there were uncountably many.

I was thinking of the generalized continuum hypothesis--the idea that for any infinite set of a given cardinal (aleph_n), its power set is always of the next higher cardinal (2^aleph_n = aleph_n+1), with no cardinals in between and no largest such cardinal. Those cardinals are obviously in one-to-one correspondence with the natural numbers; i.e., countably infinite.

You may enjoy this site, together with some of its links.

https://en.wikipedia.org/wiki/Large_cardinal

As evidence (for me) that maybe the number of transfinite cardinals is uncountable is this from the above:
"Large cardinals are understood in the context of the von Neumann universe V, which is built up by transfinitely iterating the powerset operation." Transfinite iteration is presumably meaningful. I infer that it means that there are uncountably many results.

And it occurs to me that in trying to figure out whether there are countably or merely uncountably many, we may be in the position of children at a beach concerned with a few grains of sand in the presence of an entire ocean.

The real numbers are at the center of both practice and intuition. While you have focused on Cantor, the nature of the real numbers as theoretical entities is just as strange and philosophically questionable. Why aren't you railing against the 'superstition' that 2 has a square root? Have you ever seen it? It's a theoretical construction — ee

I wouldn’t label it ‘superstition’, but an abstraction, like point, line, circle, the continuum, etc., all mental constructs for purposes of measurement. They are practical conveniences

Or as Charles Sanders Peirce aptly put it, mathematics is the science that draws necessary conclusions about purely hypothetical states of things. — aletheist

Or as Bertrand Russel said, "Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."

Ironically, there is a countably infinite number of cardinals, only the smallest of which is itself countable. — aletheist

This is not true. There's a proper class of Alephs, indexed by the proper class of ordinal numbers. After $\aleph_0, \aleph_1, \aleph_2, \dots$ come $\aleph_\omega, \aleph_{\omega + 1}$, and onward forever; too many Alephs to ever be captured in a set.

https://en.wikipedia.org/wiki/Aleph_number

There's a proper class of Alephs, indexed by the proper class of ordinal numbers. After ℵ0,ℵ1,ℵ2,… come ℵω,ℵω+1, and onward forever; too many Alephs to ever be captured in a set. — fishfry

Thanks for the correction.

I wouldn’t label it ‘superstition’, but an abstraction, like point, line, circle, the continuum, etc., all mental constructs for purposes of measurement. They are practical conveniences — sandman

OK, but then many mathematicians would agree with you. To me math is something like the formalization of intuitions. I like formalism, but it doesn't speak to the beauty many find in it. And when I and other math people I know are doing math, we need intuition to write proofs and make sense of proofs.

That said, at some points proofs become too long. No one can remember all of the steps. One has to trust the machinery of what one has proved but no longer keeps in mind. Or one uses machinery that one hasn't checked. We can switch into a playing-with-symbols mode, and sometimes we have to. In short, no philosophy of mathematics gets it quite right for me. They all focus on this or that aspect. I will say that I'm not a Platonist, since I don't think a community can see around its own eyes. Shared intuitions are plausible. More feels like reaching.

Thanks for the effort. You've made many many interesting statements in your post. I'm grateful. Thanks.

I was using the "plucked chicken" example to only point out what I think is important in math - consistency. I'm vaguely aware that we can add/delete axioms and build entirely new worlds of numbers and geometry. As must follow inconsistencies between such worlds are expected. The only example I can think of to explain this is that in non-euclidean space the sums of a triangle need not equal 180 degrees.

However, each mathematical world however constructed must be internally consistent i.e. the axioms and definitions within a given system shouldn't lead to contradictions.

In the case of the bijection, defined as each element of one set being paired with exactly one element in another set, when applied to the sets natural numbers and even numbers we certainly do arrive at the conclusion that there's a bijection between these sets.

Yet, following the same principle - pairing one element to another of these two sets - we arrive at a situation where every even number is paired with an even number in the set of natural numbers with the odd numbers left unpaired. Here we have what I think is an inconsistency - the same rule (pairing elements of one set with elements of another) producing, depending on the way you do the pairing, different, actually contradictory, results. Math can't have contradictions can it?

Here we have what I think is an inconsistency - the same rule (pairing elements of one set with elements of another) producing, depending on the way you do the pairing, different, actually contradictory, results. Math can't have contradictions can it? — TheMadFool

What you leave out, and what has apparently been left out, of all of this is that the sets have to first be well-ordered. Then the bijection is a two-way Hobson's choice: next rider, next horse. And you never run out of either riders or horses. The problem with irrationals, is that they cannot be put into a well-ordering. (I.e., whenever you put two net to each other, you can then always put one in between - actually, a whole infinity of numbers in between.)

What you leave out, and what has apparently been left out, of all of this is that the sets have to first be well-ordered. Then the bijection is a two-way Hobson's choice: next rider, next horse. And you never run out of either riders or horses. The problem with irrationals, is that they cannot be put into a well-ordering. — tim wood

I'm not sure I understand your point. Some examples that come to mind:

* The unit interval [0,1] can be bijected to the interval [0,2] but neither set is well-ordered in its usual order and both are uncountable sets full of irrationals (and some rationals too).

* Ok you say but at least in that case the obvious bijections preserve order. But how about bijecting [0,1] to the open unit interval (0,1)? Then there's a bijection but it does not preserve order.

* The naturals biject to the rationals and the rationals in their usual order are not well-ordered.

* For that matter the naturals can be bijected to the integers and the integers are not well-ordered.

* In fact any set whatsoever may be well-ordered as a consequence of the axiom of choice.

* And finally, and counterintutively, the reals might be well-ordered even in the absence of the axiom of choice.

Well-ordering doesn't have anything to do with bijections. I can't understand the point you're making.

ps -- One more item.

* Even two well-ordered sets may not be bijectable. There are uncountable well-ordered sets.

Here we have what I think is an inconsistency - the same rule (pairing elements of one set with elements of another) producing, depending on the way you do the pairing, different, actually contradictory, results. Math can't have contradictions can it? — TheMadFool

There is no contradiction. The definition of cardinal equivalence is that there exists at least one bijection between the two sets. I don't know why you have a psychological block against grokking that.

Guy robs a bank, gets caught. In the interrogation room the detective says, "Fred we know you're the bank robber." Fred says, "Oh you are wrong. Here is a list of all the banks in the state that I didn't rob. I even have a notarized statement to that effect from the manager of every single bank in the country that I did not rob."

Is Fred a bank robber? Yes of course. He robbed a bank! He robbed one single solitary bank and DIDN'T rob all the others. But he's a bank robber.

It's an existential quantification, "there exists," and not a universal one, "for all." Someone is a bank robber if they ever robbed a bank, even if there are many banks they didn't rob. Two sets are cardinally equivalent if there is a bijection between them, even if -- as must ALWAYS be the case -- there are maps between them that are not bijections.

Someone murders someone, they're a murderer. No use parading before the jury the seven billion human beings they DIDN"T murder. That lady cop in Dallas a few months ago who shot a guy sitting in his living room eating ice cream. She was convicted of murder. She's in prison as we speak, ten years if I recall. No use trying to point to all of her neighbors who she didn't kill. She killed one guy. That makes her a murderer in the eyes of the law.

Why is this simple point troubling you? If you're on the jury do you say, "Well, the prosecutor showed that she murdered someone. But she didn't murder EVERYONE." You find her not guilty on that basis? Of course not! Right?

Even Hitler didn't murder EVERYONE. You think he got a bad rap? LOL.

* The unit interval [0,1] can be bijected to the interval [0,2] but neither set is well-ordered in its usual order and both are uncountable sets full of irrationals (and some rationals too).
* Ok you say but at least in that case the obvious bijections preserve order. But how about bijecting [0,1] to the open unit interval (0,1)? Then there's a bijection but it does not preserve order. — fishfry

Nope. Google bijection.

* The naturals biject to the rationals and the rationals in their usual order are not well-ordered. — fishfry

what is their usual order? What matters is that they can be well-ordered.

Well, reading the rest of your post, let's try this. Name any two real numbers that are next to each other in a well-ordering of the set of real numbers. Granted, given two such numbers as a set of two numbers, the elements of that set can be well-ordered, but the set in question is the set of reals.

The real problem with all of this stuff is not that everyone else is wrong and you alone are correct - no danger of that. Nor even that you cannot handle it, because you can. The real difficulty with these ideas is just getting used to them. So in the words of my neighbor, a retired master sergeant, who so far has always made sense, suck it up, buttercup, and get used to them

Nope. Google bijection. — tim wood

What? There's no order-preserving bijection between [0,1] and (0,1). Yet there is most definitely a bijection But why would I need to Google bijection? Perhaps you can point to the paragraph on Google that you are referring to.

Can you please be specific in your objection?

* Do you agree that there's a bijection between [0,1] and (0,1), the closed and open unit intervals in the real numbers, respectively?

* Do you understand that such a bijection could not possibly be order-preserving?

* The naturals biject to the rationals and the rationals in their usual order are not well-ordered.
— fishfry
what is their usual order? What matters is that they can be well-ordered. — tim wood

What is the usual order on the rationals? You aren't sure? Come on, man. The usual order is the usual order.

Well, reading the re

st of your post, let's try this. Name any two real numbers that are next to each other in a well-ordering of the set of real numbers. — tim wood

Pi and 47. I see that you are not familiar with the notion of well-ordering the reals, or well-orderings in general. If you were you could not ask this question.

Granted, given two such numbers as a set of two numbers, the elements of that set can be well-ordered, but the set in question is the set of reals. — tim wood

Which can be well-ordered: definitely using the axiom of choice; and possibly even in the absence of choice.

The real problem with all of this stuff is not that everyone else is wrong and you alone are correct - no danger of that. Nor even that you cannot handle it, because you can. The real difficulty with these ideas is just getting used to them. So in the words of my neighbor, a retired master sergeant, who so far has always made sense, suck it up, buttercup, and get used to them — tim wood

Wow. Just wow. What is it with this forum? Tim, you're as wrong as wrong can be.

Of course I'm perfectly used to people on this forum being wrong about some mathematical topic and getting insulting when their ignorance is pointed out. The old forum was never this nasty but this place is. But on the mathematical facts, well you haven't presented any.

Nope. Google bijection. — tim wood

Define $f: [0,1] \rightarrow [0,2]$ by $f(x) = 2x$

Claim: $f$ is a bijection.
Proof: We will proceed by showing that $f$ is injective and surjective.

Injective subproof:
$f$ is injective when and only when for all $x,y$, $f(x) = f(y)$ implies $x=y$. Let $x,y \in [0,1]$ be arbitrary, then assume (for conditional proof) that $f(x)=f(y)$, by the definition of $f$ this implies $2x=2y$ which in turn implies $x=y$. So $f$ is injective.

Surjective subproof:
$f$ is surjective when and only when for all $y \in [0,2]$ there exists $x \in [0,1]$ such that $y = f(x)$. Let $y$ in $[0,2]$ be arbitrary, then consider $x=y/2$. Firstly, this is in $[0,1]$ because the minimum of $x$ is $0$ and the maximum is $1$. Then we evaluate $f(x) = f(y/2) = 2y/2 = y$. Since $y$ was arbitrary, $f$ is surjective.

Therefore $f$ is injective and surjective. By definition, $f$ is bijective.

Two sets are of equal cardinality when and only when a bijection exists between them. $f$ exists (it is a well defined function (proof omitted) between $[0,1]$ and $[0,2]$). Therefore $[0,1]$ and $[0,2]$ are of equal cardinality.

Regarding the order preserving bit. This function is total order preserving because it's monotonic increasing. But the definition of bijection or being of equal cardinality does not require order preservation. There are easy finite set examples for this. An infinite set example is $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ with $f(x) = 1/x$ - this one is a bijection that reverses inequalities (it's monotonic decreasing).

Neither of these say anything about well orders; the order above is the standard total order on the reals, rather than any well ordering. The ordering has sets which do not have a least element. (Real) sets with a least element contain their greatest lower bound (which is their set minimum) - and an interval like $(0,1)$'s greatest lower bound is $0$, which isn't in the set by definition. So the standard order on the reals isn't a well order; because there's at least one subset of it which does not have a least element.

herefore [0,1] and [0,2] are of equal cardinality. — fdrake

Wait ... @tim wood was confused about THAT?? I didn't mention it because it's so obvious, at least to anyone who would jump into a discussion of bijections. Thanks for pointing out that I gave @Tim too much credit. I thought he was confused about the far more tricky bijection between [0,1] and (0,1).

It's just in case. I think you need to check your math privilege. :yum:

"Can you even imagine? They don't even understand Ito integrals..." (paraphrased discussion excerpt from the staff room)

I think you need to check your math privilege. — fdrake

I just don't understand the insult culture around here. Over the past couple of years I've had to take extended breaks from this forum because someone started piling on personal insults at me over technical matters on which they happened to be flat out wrong. Not because I can't snap back; but because I'm perfectly capable of snapping back, and that's not what I'm here for. I'd suggest to members that whenever they throw an insult in lieu of a fact, perhaps they should consider whether they've got any facts.

I just don't understand the insult culture around here, especially involving technical matters. — fishfry

For me it can be difficult to tell whether something's technical or not if I'm unfamiliar with it. Knowing what's relevant to what in a technical matter is part of getting to know the technical matter. In the absence of that information, or a sufficiently well explained link between the technical matter and the discussion topic, it can feel extremely patronising. I think there's a hierarchy here.

(1) Basic math literacy to general public: "They can't multiply fractions, can you even imagine!"
(2) Math education to basic math literacy: "They can't do calculus, can you even imagine!"
(3) Undergrad math education to math education: "They can't do the Gram-Schmidt process, can you even imagine!"
(4) Graduate math education to undergrad math education becomes field specific.
(5) Research math to graduate math becomes subfield specific.