Sorry, k_i ∈ (0, 1) was supposed to mean that k_i is in a set consisting of 0 and 1. Not sure what the correct notation should be.

:up: Looks good to me. I've never even thought of using numerical methods in proofs! That's very cool.



Think it's curly braces. $\{\}$ They need \ in front of them in math mode.

Just to be clear I will reiterate the proof in slightly more detail

We will map the number 5.7 ... — Umonsarmon

Note that this is rational number; more specifically, a rational number whose proper-form denominator is equal to (2^n)*(5*m) for integers n and m. The important characteristic is that there is a finite number of digits on each side of the decimal point.

... For arguments sake we will say the line terminates here at point E

Now I measure the distance from A to E ... This distance will be some multiple of a 1/2 x some a/b

But what if the process does not terminate? Then this won't always be true.

If I understand your poorly-described algorithm, what you really get is a terminating sum that looks like this, for a set of increasing integers {n1,n2,n3,...,nk}:

2^(-n1)+2^(-n2)+2^(-n3)+...2^(-nk)

If that isn't exactly right, it's something similar. Yes, this is a rational number.

But if the set of integers does not end, it does not have to be rational. And if the number you trace is irrational, your algorithm will not terminate. So no, you did not disprove Cantor.

+++++

A lot of people feel there is something wrong with Cantor's Diagonal Argument. That's because what they were taught was an invalid version of CDA. Some of the issues are just semantics (ex: it didn't use real numbers), and some are correctable (if you use numbers, you need to account for real numbers that have two decimal representations).

But one invalidates what they were taught as a proof. The correct proof is right, though. Here's a rough outline:

• Let t represent any infinite-length binary string. Examples are "1111....", "0000...", and "101010...".
• Let T be the set of all such strings t.
• Let S be any subset of T that can be put into the list s1, s2, s3, ... .
• Use diagonalization to construct a string s0 that is in T but is not in S.

This proves that every listable portion of T is not all of T.

Most of the times CDA is taught, it is assumed that all of T is put into a list. Then, by proving that there is an element that is not in the list, it contradicts that you have a complete list. Thus proving by contradiction that all of T cannot be listed.

This is an invalid proof by contradiction. You have to use all aspects of the assumption to derive the contradiction. The derivation does not use the assumption of a complete list, so it cannot disprove completeness by contradiction. But it did disprove it directly.

What seems to have started the confusion was the second part of Cantor's version. If S=T, then s0 both is, and is not, an element of T. This is a proper proof by contradiction.

Thank you for this. It's a gift to be taught something. I appreciate it.

What seems to have started the confusion was the second part of Cantor's version. If a complete list is possible, it contradicts the proven fact that every list is incomplete. So a complete list is impossible — JeffJo

I think most people understand the reductio ad absurdum proof. What the big problem is what then?

You see there is an inherent structural difference between giving a direct proof and having only the possibility of giving an indirect proof. What can you say about mathematical objects, that can only be shown to exist, to be true, by an indirect proof? This is where critique about Cantor really lies. Can you just assume "larger" infinities, if you indirectly prove N < R? Everything fine and dandy after that and you can assume aleph-2, aleph3, aleph4...?

I think most people understand the reductio ad absurdum proof. What the big problem is what then? — ssu

The problem is that CDA isn't a reductio ad absurdum proof; at least not as people think. The common presentation of it as reductio fails logically. As I described.

It also wasn't intended to be the focus of Cantor;s effort. It was a demonstration of the principle with an explicit set. What is known as Cantor's Theorem uses an abstract set, and proves that its power set must have a greater cardinality. (And, btw, it is a correct use of reductio.)

And we don't prove existence. Axioms do. The Axiom of Infinity establishes that the set of all natural numbers exists. The Axiom of Power Set says the power set of any existing set also exists. With those axioms, Cantor's Theorem proves that an infinite number of Alephs exist.

This is a mostly geometric argument and it goes like this. — Umonsarmon

The problem with your algorithm is that the binary representation of a natural number is finite. Therefore, your algorithm picks out only those numbers that can be expressed through finite bisections of an interval. It turns out that the unit interval is isomorphic to the set of infinite binary sequences. The diagonalization argument establishes that the set of finite binary sequences does not surject the set of infinite binary sequences. It therefore deserves the name "uncountable." However, because there is an injection from the set of finite binary sequences to the set of infinite binary sequences that does not surject the set of infinite binary sequences, this "uncountable" set is strictly larger than the "countable set," despite the obvious fact that both sets are infinite.

And we don't prove existence. Axioms do. — JeffJo

I wouldn't agree on this. Axioms don't give proofs. Perhaps we are just thinking of this a bit differently.

A proof shows that something is true. If a mathematical object, a hypothesis, theorem, lemma is shown to be true, hence it is said to exist in the mathematical realm. It's hard to argue that something is illogical or false in math, if someone has given a logical proof for it. Axioms on the other hand are just given: "a statement or proposition which is regarded as being established, accepted, or self-evidently true." Or defined in a more mathematical way:

An axiom is a proposition regarded as self-evidently true without proof. The word "axiom" is a slightly archaic synonym for postulate. Compare conjecture or hypothesis, both of which connote apparently true but not self-evident statements.

Mathworld Wolfram

I wouldn't agree on this. Axioms don't give proofs. Perhaps we are just thinking of this a bit differently. — ssu

I'm sorry, I worded that poorly.

We don't establish the existence of these sets by proof. We do it by the axioms we choose to accept. And since all proofs in mathematics are based on such axioms, there is no difference in the validity of either method.

Yeah well, it can happen that something that we have taken as an axiom isn't actually true.

Just look at how much debate here in this forum there is about infinity. Axiom of Infinity is anything but established and self-evidently true. The discussion here ought to show it. We just don't know yet! Bizarre to think that there are these gaping holes in our understanding of math, but they are there.

Axiom of Infinity is anything but established and self-evidently true. — ssu

And yet it sounds so simple. A set is a collection of "elements" or objects and the axiom says we can consider the collection itself as an object, but perhaps not the same kind of object - but wait, can a set be considered an element of itself??? :roll:

but wait, can a set be considered an element of itself??? — John Gill

If I remember correctly, some use that as a definition of infinity:

A set is infinite if and only if it is equivalent to one of its proper subsets.

Yeah well, it can happen that something that we have taken as an axiom isn't actually true. — ssu

With all due respect, if you want "actual truth", then you do not understand the purpose of an axiom in mathematics. The point is that mathematics contains no concept of actual truth. We define different sets of "truths" that we accept without justification. A set is invalid only if it leads to internal inconsistency, not if you think it violates an "actual truth" that is not in that set.

If we accept as true that there exists a set that contains the number 1 and, if it contains the number n then it also contains the number n+1? Then we can prove the existence of the cardinalities we name aleph0, alpeh1, aleph2, etc. Any inconsistencies you may think you have found are due to things you assume are actual truth but are not included in this mathematics.

Just look at how much debate here in this forum there is about infinity. Axiom of Infinity is anything but established and self-evidently true. The discussion here ought to show it. We just don't know yet! Bizarre to think that there are these gaping holes in our understanding of math, but they are there.

What is bizarre is that some don't understand that everything proven in mathematics is based on unproven, and unproveable, axioms.

If we accept as true that there exists a set that contains the number 1 and, if it contains the number n then it also contains the number n+1? Then we can prove the existence of the cardinalities we name aleph0, alpeh1, aleph2, etc — JeffJo

See the argument given in this OP:

https://thephilosophyforum.com/discussion/7309/whenhow-does-infinity-become-infinite/p1

IE induction leads to the conclusion that aleph0 must be finite.

IMO, the axiom of infinity is nonsensical and leads to absurdities like Galileo's paradox and Hilbert's hotel.

With all due respect, if you want "actual truth", then you do not understand the purpose of an axiom in mathematics. The point is that mathematics contains no concept of actual truth. — JeffJo

My point was that axioms can be possibly false. Our understanding can change. Best example of this was that until some Greeks found it not to be true, people earlier thought that all numbers are rational. Yet once when you prove there are irrational numbers, then the 'axiom' of all numbers being rational is shown not to be true.

And not all axioms are self evident. Just look at what Devans99 wrote about the axiom of infinity above

My point was that axioms can be possibly false. — ssu

My point is that they can't. That's why they are axioms.

There are no absolute truths in Mathematics, only the concepts we choose to accept as true. While we can find that a set of axioms is not self-consistent, that does not make any of them untrue.

Best example is from Euclid, who first proposed an axiomatic mathematics. He thought that he should be able to prove a self-evident fact from his axioms in plane geometry. That given a line and a point not on that line, then there must exist exactly one line in the plane they define that passes through the point and is parallel to the line. It turns out that this needs to be another axiom. And that you can define consistent geometries if that axiom says there is one, many, or none.

Axioms don't need to be self-evident; in fact, that's what requires them to be axioms.

The normal definition of axiom:

'a statement or proposition which is regarded as being established, accepted, or self-evidently true.'

The mathematical definition of an axiom:

'a statement or proposition on which an abstractly defined structure is based'

The definition of a mathematical statement:

'A meaningful composition of words which can be considered either true or false'

So an axiom can be false; invaliding any results derived from that axiom.

?

My point is that they can't. That's why they are axioms. — JeffJo

I already gave an example of what was thought to be an axiom that wasn't. Greeks thought that all numbers were commensurable. The thought was for them self evident: math was so beautiful and harmonious. Yet all numbers weren't commensurable.There exists irrational numbers (and even transcendental numbers). What we had accepted to be true wasn't the case.

As Reuben Hersh says: "Mathematical knowledge isn’t infallible. Like science, mathematics can advance by making mistakes, correcting and recorrecting them."

Also a good point!

but wait, can a set be considered an element of itself??? :roll: — John Gill

Mathematical set theory and foundations provide a playground for combative metaphysicians. And if you contemplate my question you might see why.

My point was that axioms can be possibly false. Our understanding can change. Best example of this was that until some Greeks found it not to be true, people earlier thought that all numbers are rational. Yet once when you prove there are irrational numbers, then the 'axiom' of all numbers being rational is shown not to be true. — ssu

You put 'axiom' in inverted commas for good reason, even if you didn't understand it. That all numbers are rational wasn't an axiom - it was a definition, an informal intuition, or a conjecture, depending on how they approached numbers in their thinking. And accepting or rejecting an axiom does not amount to judging it to be true or false; as has been repeatedly explained to you, that doesn't even make any sense.

I already gave an example of what was thought to be an axiom that wasn't. — ssu

You gave an example of a near-religious belief. It was never an axiom in a consistent mathematics.This is similar to the belief that we can't treat aleph0 as a valid mathematical concept.

And I gave you an example where three different, consistent mathematics have been created based on three different axioms that contradict each other. The point is that neither mathematics, nor the axioms any specific field of mathematics is based on, are intended to represent absolute truth. You cannot win this debate with a nonsense claim like "the axiom is wrong."

You put 'axiom' in inverted commas for good reason, even if you didn't understand it. That all numbers are rational wasn't an axiom - it was a definition, an informal intuition, or a conjecture, depending on how they approached numbers in their thinking. — SophistiCat

Your reasoning of it being an conjecture or an informal intuition can be done only in hindsight. The definition of an axiom is "A self evident proposition requiring no formal demonstration to prove its truth, but received and assented
to as soon as mentioned". People were thinking about numbers and their commensurability exactly like as an axiom: a self evident proposition requiring no demonstration, something that was evident. Before it wasn't. And now it's an 'informal intuition'. Math simply is similar to science: we can make mistakes.

And I gave you an example where three different, consistent mathematics have been created based on three different axioms that contradict each other. The point is that neither mathematics, nor the axioms any specific field of mathematics is based on, are intended to represent absolute truth. You cannot win this debate with a nonsense claim like "the axiom is wrong." — JeffJo

Yet what you are stating is a philosophical view of mathematics. What you are basically saying is that: "You cannot win this debate because you don't accept formalism!"

Formalism goes exactly on these lines you said above: Math isn't a body of propositions representing an abstract sector of reality, but more akin to a game. Nothing to do with ontology of objects or properties and something more like chess. The truths expressed in logic and mathematics tell us hardly anything about numbers or sets themselves. It's just basically a game where axioms are just premises that define the system used and how useful the system is or isn't doesn't matter. Hence if you have a math system where 0=1, then it means just that anything goes, right? That's formalism.

And lastly, I'm not here to win anything, but to have a conversation from which I hopefully can learn something.

An axiom is a statement - statements are true or false. End of story.

Treating aleph0 as a mathematical concept leads to paradoxes, eg:

https://en.wikipedia.org/wiki/List_of_paradoxes#Infinity_and_infinitesimals

What is a paradox? It's a contradiction. So a paradox indicates the presence of an reductio ad absurdum argument - IE we have made a false assumption (one of our axioms is false) and it has lead to an absurd conclusion.

In the examples linked above, that false assumption is the axiom of infinity.

The actually infinite is just a mental convenience - it is merely economical mentally to talk about the number of reals in [0,1] as being actually infinite. Of course in reality, it is no such thing - it is an example of potential infinity - those numbers do not take on real existence until we list them (actualise them).

So the axiom of infinity works in our minds, but does not work in any sort of reality - hence all the paradoxes.

Many things work in our minds (square circles, talking trees, levitation) but have no basis in reality. Actual infinity is one such thing.

The definition of an axiom is "A self evident proposition requiring no formal demonstration to prove its truth, but received and assented
to as soon as mentioned" — ssu

That's not the definition of an axiom, as you ought to have learned by now if you were paying attention.

Seems like then you have your your own definition...

Definitons of axiom:

"An axiom is a proposition regarded as self-evidently true without proof. The word "axiom" is a slightly archaic synonym for postulate. Compare conjecture or hypothesis, both of which connote apparently true but not self-evident statements." Axiom - Wolfram Mathworld

"An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. " Axiom - Wikipedia

"1) a statement accepted as true as the basis for argument or inference. 2) an established rule or principle or a self-evident truth" axiom - Merriam Webster dictionary

Seems like then you have your your own definition — ssu

No, seems like you are only interested in playing dictionary games. You can join the other idiot then, I am not interested.

Name calling does not win arguments.

An axiom is a statement - statements are true or false. End of story. — Devans99

Not the end of the story. Definitions are not commutative. An axiom is indeed a "proposition regarded as self-evidently true without proof." That does not mean that every "proposition regarded as self-evidently true without proof" is an axiom.

Looking further at Wolfram, an[ i]Axiomatic System[/i] is a "logical system which possesses an explicitly stated set of axioms from which theorems can be derived." So once again, the statement you claimed was an axiom was never stated as part of such a set, from which theorems could be derived. It was not an axiom, it was a near-religious belief.

Yet what you are stating is a philosophical view of mathematics.

No. What I am saying is that theorems in a field of mathematics need to be based on some set of accepted truths that are called the axioms of that field. Such a set can be demonstrated to be invalid as a set by deriving a contradiction from then, but not by comparing them to other so-called "truths" that you choose to call "self-evident."

What I am explicitly saying is that any arguments not based in the axiomatic system of a field of mathematics cannot not say anything about that field.

Spot on.

A couple of months ago the forum was infested with bad theology. Now it's bad maths.

Well, you argued that I gave an incorrect definition, SophistryCat.

Besides, one shouldn't assume that one school of Mathematical philosophy is correct and another is not. I gather that JeffJo thinks on the line of mathematical formalism when it comes to axioms (I assume, of course I could be wrong) while you've just said that I ought to know better...besides being an idiot.