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How many real numbers exist? New proof moves closer to an answer

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Re: How many real numbers exist? New proof moves closer to an answer

#91

Post by kmill »

tgflynn wrote: post #72
rssoconnor wrote: post #65

Earlier quoted context omitted.

> but at the same time there is no "right answer"—all of these axioms, after all, are independent of ZFC, and it's "ok" to add any of them. Minor quible: Just because a potential axiom is independent of ZF(C) doesn't make it necessarily "okay" to add. Potential axioms can be unsound, for example if they prove new / untrue Sigma_1 statements. As an example, even in the likely circumstance that ¬Con(ZFC) is independent…

> he resulting system is unsound (in the sense of Tarski) because it asserts the existence of natural numbers that have no "written form" (i.e. the existance of natural numbers that are larger than any term you can write to denote a natural number). Isn't that basically the definition of the natural numbers, ie. if you write down any natural number (say n) I can always construct a natural number that is larger than i…

This surprisingly doesn't mean repeatedly adding 1 will exhaust all natural numbers -- there are models for the natural numbers with elements that can't be reached this way!

The ultrafilter construction gives one such model. You take the set of all sequences of natural numbers (n1, n2, n3, ...) then use an ultrafilter to decide which of these sequences are considered to be equal. The usual operations of natural numbers are defined term-by-term. You can think of the usual natural numbers as being the sequences (0,0,0,...), (1,1,1,...), (2,2,2,...) and so on. However, there are many additional numbers in this system, like (0,1,2,...) that are strictly greater than all the usual numbers, and these cannot be reached by repeatedly adding one. (The reason (0,1,2,...) is greater than (n,n,n,...) is that if we do the comparison term-by-term we get (false,false,...,false,true,true,true,...), and the ultrafilter will decide the comparison is true because it is true for all but finitely many terms. Ultrafilters are devices to consistently turn infinite sequences of trues and falses into a single decision, but every ultrafilter will make a true decision in the case there are only finitely many falses.)

Counter-intuitively, proofs by induction still work for this system... speaking with no authority here, maybe an intuition is that it's doing a hypercomputation.

Re: How many real numbers exist? New proof moves closer to an answer

#92

Post by rssoconnor »

tgflynn wrote: post #72
rssoconnor wrote: post #65

Earlier quoted context omitted.

> but at the same time there is no "right answer"—all of these axioms, after all, are independent of ZFC, and it's "ok" to add any of them. Minor quible: Just because a potential axiom is independent of ZF(C) doesn't make it necessarily "okay" to add. Potential axioms can be unsound, for example if they prove new / untrue Sigma_1 statements. As an example, even in the likely circumstance that ¬Con(ZFC) is independent…

> he resulting system is unsound (in the sense of Tarski) because it asserts the existence of natural numbers that have no "written form" (i.e. the existance of natural numbers that are larger than any term you can write to denote a natural number). Isn't that basically the definition of the natural numbers, ie. if you write down any natural number (say n) I can always construct a natural number that is larger than i…

Dropping down to Peano Arithmetic for a moment. We can consider adding a new constant 'c' for a natural number to the language along with the following infinite list of axioms about this remarkable constant:

- 0 - 1 - 2 - 3 ...

Adding all these axioms is consistent. I.e. you can do induction upto 'c', whatever it is. Why is it consistent? Because if there was a contradiction, the proof of such a contradiction would be finite, and hence can only use a finite number of these new axioms (this is a so-call compactness argument). But clearly any finite subset of this list of axioms is consistent because it has a model where c is just defined to be 1 more than the largest numeral appearing in that list.

But all those infinite number of axioms taken together creates an unsound system because it claims that 'c' denotes a natural number that is larger than every written numeral.

Heading back to ZFC land, it turns out that (assuming ¬Con(ZFC) is independent of ZFC) adding ¬Con(ZFC) to ZFC similarly is similarly unsound in that it yields only models that have elements that are larger than every written numeral.

Re: How many real numbers exist? New proof moves closer to an answer

#93

Post by hypersoar »

tgflynn wrote: post #72
rssoconnor wrote: post #65

Earlier quoted context omitted.

> but at the same time there is no "right answer"—all of these axioms, after all, are independent of ZFC, and it's "ok" to add any of them. Minor quible: Just because a potential axiom is independent of ZF(C) doesn't make it necessarily "okay" to add. Potential axioms can be unsound, for example if they prove new / untrue Sigma_1 statements. As an example, even in the likely circumstance that ¬Con(ZFC) is independent…

> he resulting system is unsound (in the sense of Tarski) because it asserts the existence of natural numbers that have no "written form" (i.e. the existance of natural numbers that are larger than any term you can write to denote a natural number). Isn't that basically the definition of the natural numbers, ie. if you write down any natural number (say n) I can always construct a natural number that is larger than i…

You can find a natural number that is bigger than any one natural number, but you can't write one down that's bigger than every natural number in the ZFC sense.

Re: How many real numbers exist? New proof moves closer to an answer

#94

Post by croes »

morpheos137 wrote: post #25
croes wrote: post #22

Earlier quoted context omitted.

You are confusing numbers with natural numbers. "A number is a mathematical object used to count, measure, and label."

You can't count reals or complex numbers. Real numbers by definition describe objects with infinite precision. In the real world infinite precision cannot exist therefore real numbers are the limit of an arbitrary precision measuring process not real themselves. Meanwhile natural numbers like "two" most certainly do exist as a quantitative attribute of a set. Show me how you can count with reals.

I don't need to, I can measure and label with real numbers. That's enough for being a number.

Re: How many real numbers exist? New proof moves closer to an answer

#95

Post by dogecoinbase »

lupire wrote: post #90
rssoconnor wrote: post #65

Earlier quoted context omitted.

> but at the same time there is no "right answer"—all of these axioms, after all, are independent of ZFC, and it's "ok" to add any of them. Minor quible: Just because a potential axiom is independent of ZF(C) doesn't make it necessarily "okay" to add. Potential axioms can be unsound, for example if they prove new / untrue Sigma_1 statements. As an example, even in the likely circumstance that ¬Con(ZFC) is independent…

why is that "unsound"? what's wrong with an unwritable natural? Almost all reals are unwritable.

Soundness in this sense (sigma_1 soundness) is defined by equivalency to the standard model (specifically, all sentences provable in the system must be provable in the standard model).

Re: How many real numbers exist? New proof moves closer to an answer

#96

Post by Sharlin »

sillysaurusx wrote: post #69
svat wrote: post #57

Earlier quoted context omitted.

No amount of empirical data is ever "good enough" for the mathematical standard of proof. (It may be enough for mathematicians to "believe" something in some informal sense, but not enough to consider it "proved".) You can see some examples at the answers to these questions; maybe at least one of them will be interesting to you: - https://math.stackexchange.com/questions/514/conjectures-tha... - https://math.stackexc…

That was another question in the back of my mind -- famously, the four color theorem was proved by computers through exhaustive analysis (checking every possibility). At the time, it was controversial as a "proof" since it didn't really take the usual form of a proof. I've often wondered "Why can't we do something like that, but for all instances of things like the Collatz conjecture?" Of course, it's computationally…

Math is simply not a science, not in the sense of "natural science". Math research does not follow the scientific method, it is not an iterative effort to construct (wrong, but useful) models of some external universe like science does. Math is a thing of its own, and it's best just to not compare it to physics or biology or gender studies or whatever.

Re: How many real numbers exist? New proof moves closer to an answer

#97

Post by TuringTest »

morpheos137 wrote: post #25
croes wrote: post #22

Earlier quoted context omitted.

You are confusing numbers with natural numbers. "A number is a mathematical object used to count, measure, and label."

You can't count reals or complex numbers. Real numbers by definition describe objects with infinite precision. In the real world infinite precision cannot exist therefore real numbers are the limit of an arbitrary precision measuring process not real themselves. Meanwhile natural numbers like "two" most certainly do exist as a quantitative attribute of a set. Show me how you can count with reals.

At the end of the day, a number is a _reification_ (or thing-a-fication, watching a process as a separate entity) of the processes of mapping and ordering. You may begin with a very simple use of those processes, which get you the natural numbers. But you may use them in more creative ways, which will get you other different classes of numbers. Mathematician love to explore all the implications of using basic processes and combining already defined numbers to create new kinds, never seen before.

For example, if you start with number '1' and apply operator _successor_ (or "adding one more"), you get the *natural numbers*, which are a mapping from the size of sets to strings of operators {'1', 'successor(1)', 'successor(successor(1))', ... }. You can use this process to define an order: number A is smaller than B if you can repeatedly apply _successor_ to A and generate B in a finite time. (Not the best definition, I know, but bear with me for a second).

If you reverse the _successor_ operator, you get the _predecessor_, which can be used to dismount large numbers and make them smaller. If you apply _predecessor_ to '1', you get 'predecessor(1)' which doesn't match any of the *natural* numbers as defined in the paragraph above. However, this new number is useful because you can map it to a collection without elements, allowing you to count a set of size 'zero'. This set doesn't exist, but it has a well defined size thanks to the process explained above. (You may as well apply _predecessor_ to 'zero' and count *negative numbers*, which allow you to create a mapping with debt, and thus count _I owe you_ amounts that don't exist in the physical world either).

Now as for real and complex numbers, they can't count physical objects as you said because at some point you can't measure smaller and smaller magnitudes (though you can create orders with them, even if it doesn't directly involve the _successor_ operator). But mathematicians use them to count the size of _infinite sets_, which being immaterial never lose precision; you just know you can repeat their defining processes once and again, showing that there exist a mapping between any step in the process and the instance of the real or complex number that correspond to its size.

Re: How many real numbers exist? New proof moves closer to an answer

#98

Post by lupire »

gorgoiler wrote: post #43
jaywalk wrote: post #8

I'm not trying to be flippant, although it may come off that way: why does any of this matter?

We don’t know but looking into the past hints at the future. Cantor, Hilbert and Gödel gave us Church who gave us Turing. Turing and Flowers gave us the machines as well as the theory. All of them put together gave us type systems and types are how you formally prove that your 747 software is free of, if not all bugs, then at least certain large classes of error. There is a clear line of connections from Cantor (1890…

Eh, this argument kind of falls apart with higher infinities, where the numbers you are studying are bigger than the power set of all interactions among all particles in all possible universes.

Re: How many real numbers exist? New proof moves closer to an answer

#99

Post by Ivoirians »

schoen wrote: post #81
Ivoirians wrote: post #60

Earlier quoted context omitted.

They've verified that the Collatz conjecture holds for all numbers up to ~2^68, but that's precisely 0% of all the numbers that need to be checked. But more importantly, the goal of (pure) mathematics isn't to declare truths. If you had a machine from God himself that outputted True or False for theorems you put in, that wouldn't demotivate (pure) mathematicians from doing the work they're doing. Understanding the re…

Though note that that True/False machine could also quickly lead to the discovery of actual proofs, for example by making it extremely efficient to search for counterexamples. "The Collatz conjecture is true for all n ≤ some_huge_number." Also perhaps by making it extremely efficient to search for correct proofs in a lexicographically ordered list of proofs. "The shortest valid proof of the Collatz conjecture in the…

This is true, a universal oracle would probably lead to some chaos in the world's computer science departments. I wonder how many mathematicians would switch over to help formalize and solve their fields, and how many would stick to their pencils and paper :P

Re: How many real numbers exist? New proof moves closer to an answer

#100

Post by TuringTest »

HelloNurse wrote: post #56
morpheos137 wrote: post #25

Earlier quoted context omitted.

You can't count reals or complex numbers. Real numbers by definition describe objects with infinite precision. In the real world infinite precision cannot exist therefore real numbers are the limit of an arbitrary precision measuring process not real themselves. Meanwhile natural numbers like "two" most certainly do exist as a quantitative attribute of a set. Show me how you can count with reals.

In practical terms, "counting and measuring" means well-behaved arithmetic operations like addition, subtraction, multiplication, division, roots etc. (and often specific algebraic structures like rings, fields etc.) Rational and real numbers represent the most intuitive concept of quantity with different cardinality; natural numbers are more basic in theory but a restricted special case in most application (they can…

> The same applies to rational numbers (which exist more than real numbers)

Wait, did you say there are more rationals than reals? Isn't that the other way around?

I don't know if that's a slip of the tongue or I'm missing something in my recall of basic math lessons

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