3 ms·

> We assume, for example, that there is always a number following another number Assuming you're referring to natural numbers or integers, that's not an assump

> We assume, for example, that there is always a number following another number Assuming you're referring to natural numbers or integers, that's not an assumption: https://proofwiki.org/wiki/Natural_Numbers_are_Infinite https://proofwiki.org/wiki/Natural_Numbers_are_Infinite

ducttapecrown 3y ago
The existence of a set with infinite cardinality is one of the axioms of ZFC. https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory#7._Axiom_of_infinity https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...
anon291 3y ago
The statement 'there are infinitely many natural numbers' is not the same as 'there's a natural number following every other natural number'. In particular, the real numbers are infinite, but there is no unique real number following another real. Moreover, the natural numbers are typically defined axiomatically, either directly or via a set-based representation. Either way, you run into the same issue which is that eventually you arrive at a set that, due to known physical constraints of the universe, is impossible to identify and you must reasonably ask yourself if such a number exists.
- denton-scratch 3y ago
  Real numbers != Natural numbers. Every natural number has a 'next'; there is no 'next' for a real number.
  - anon291 3y ago
    > Every natural number has a 'next' Again... an axiomatic statement that is not well defined. What does it mean for something to have a 'next'. Shouldn't it be the case that if something has a next, then it can be named, identified, and perhaps even written down in some manner? Yet, by the same axioms, there certainly exist natural numbers that we cannot write down simply because there are not enough atoms in the universe that could be used to write them down on or with. So basically, we have a conundrum, we say something exists after some other, yet for sure such a thing cannot be identified in any meaningful way, and its existence is just some conjecture that can never be proven. In what way is such a number distinguishable from any of the other infinite numbers that are supposedly greater than it? Since none can be written and all we can really say is that it's greater than whatever other number we have, one again questions whether or not the statement 'every natural number has a next' is truly well defined.
    - denton-scratch 3y ago
      > So basically, we have a conundrum, we say something exists after some other, yet for sure such a thing cannot be identified in any meaningful way, and its existence is just some conjecture that can never be proven. Not being able to write a number down doesn't make it's existence a matter of conjecture. You could say the same thing about Pi.
      - anon291 3y ago
        > Not being able to write a number down doesn't make it's existence a matter of conjecture. You could say the same thing about Pi. Depending on your philosophy of mathematics, there is good reason to believe that some real numbers do not exist. In particular, pi is a computable number, but many reals are not. Thus, we end up with a place where we conjecture that certain things exist yet simultaneously say there's (1) no way to write it down and (2) moreover, there's no systematic way to describe it. Given that with pi, there are many programs that given an N, can compute pi to N many digits, I do think it's reasonable to say that pi can be identified. But, there are infinitely many numbers that cannot. In fact, the vast majority of real numbers that supposedly exist cannot be computed to any arbitrary precision with a turing machine. Thus, they cannot be identified. The general term for this philosophical approach towards mathematics is mathematical nominalism. What I'm seeing in this thread though is an implicit assumption that nominalism is false, despite being unaware of this assumption. I believe these sorts of hidden biases are dangerous. While I don't necessarily subscribe to nominalism, I think it's worth consideration, and I do think it brings up several interesting questions that cannot simply be ignored because 'well I believe it exists'. References: https://plato.stanford.edu/entries/nominalism-mathematics/ https://plato.stanford.edu/entries/nominalism-mathematics/ More interesting reading: https://philosophy.stackexchange.com/questions/81414/if-most-numbers-are-uncomputable-in-what-sense-do-they-exist https://philosophy.stackexchange.com/questions/81414/if-most... https://math.stackexchange.com/questions/4322297/in-what-sense-does-a-number-exist-if-it-is-proven-to-be-uncomputable https://math.stackexchange.com/questions/4322297/in-what-sen... Many automated theorem provers can only prove things by construction, thus computability is the requirement for 'existence' in these systems (calculus of constructions via Coq, LEAN, etc). In other words, they follow a constructivist approach to mathematics, which is rather interesting as such approaches require us to elide a lot of 'obvious' axioms we take for granted (such as the law of excluded middle). Several things shake out of this approach such as the conclusion that all functions are continuous. (https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_mathematics) https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_...)
roywiggins 3y ago
That proof assumes that "s(n)=n+1 is clearly an injection" - ie that every natural number has a unique successor, and uses it to prove something about the size of the set. But it doesn't prove there is "always a number following another number", it assumes it. imho it's part of the definition of what the set of natural numbers even is. Peano arithmetic just has S(n) always exists and is injective as axioms.
- anon291 3y ago
  Correct. All descriptions of the natural numbers of which I'm aware require the assumption that one can take some object and apply some operation 'S' on that object to produce another object that is neither the original object nor any of the objects produced before by repeated applications of 'S' to some root object. Whether you simply define a number as 0 and S(x) where x is a natural number or you define natural numbers as nested sets in ZFC or as church-encoded lambda terms, etc, you assume the existence of such a transformation.

ducttapecrown 3y ago

The existence of a set with infinite cardinality is one of the axioms of ZFC. https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory#7._Axiom_of_infinity https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...

anon291 3y ago

The statement 'there are infinitely many natural numbers' is not the same as 'there's a natural number following every other natural number'. In particular, the real numbers are infinite, but there is no unique real number following another real. Moreover, the natural numbers are typically defined axiomatically, either directly or via a set-based representation. Either way, you run into the same issue which is that eventually you arrive at a set that, due to known physical constraints of the universe, is impossible to identify and you must reasonably ask yourself if such a number exists.

denton-scratch 3y ago
Real numbers != Natural numbers. Every natural number has a 'next'; there is no 'next' for a real number.
- anon291 3y ago
  > Every natural number has a 'next' Again... an axiomatic statement that is not well defined. What does it mean for something to have a 'next'. Shouldn't it be the case that if something has a next, then it can be named, identified, and perhaps even written down in some manner? Yet, by the same axioms, there certainly exist natural numbers that we cannot write down simply because there are not enough atoms in the universe that could be used to write them down on or with. So basically, we have a conundrum, we say something exists after some other, yet for sure such a thing cannot be identified in any meaningful way, and its existence is just some conjecture that can never be proven. In what way is such a number distinguishable from any of the other infinite numbers that are supposedly greater than it? Since none can be written and all we can really say is that it's greater than whatever other number we have, one again questions whether or not the statement 'every natural number has a next' is truly well defined.
  - denton-scratch 3y ago
    > So basically, we have a conundrum, we say something exists after some other, yet for sure such a thing cannot be identified in any meaningful way, and its existence is just some conjecture that can never be proven. Not being able to write a number down doesn't make it's existence a matter of conjecture. You could say the same thing about Pi.
    - anon291 3y ago
      > Not being able to write a number down doesn't make it's existence a matter of conjecture. You could say the same thing about Pi. Depending on your philosophy of mathematics, there is good reason to believe that some real numbers do not exist. In particular, pi is a computable number, but many reals are not. Thus, we end up with a place where we conjecture that certain things exist yet simultaneously say there's (1) no way to write it down and (2) moreover, there's no systematic way to describe it. Given that with pi, there are many programs that given an N, can compute pi to N many digits, I do think it's reasonable to say that pi can be identified. But, there are infinitely many numbers that cannot. In fact, the vast majority of real numbers that supposedly exist cannot be computed to any arbitrary precision with a turing machine. Thus, they cannot be identified. The general term for this philosophical approach towards mathematics is mathematical nominalism. What I'm seeing in this thread though is an implicit assumption that nominalism is false, despite being unaware of this assumption. I believe these sorts of hidden biases are dangerous. While I don't necessarily subscribe to nominalism, I think it's worth consideration, and I do think it brings up several interesting questions that cannot simply be ignored because 'well I believe it exists'. References: https://plato.stanford.edu/entries/nominalism-mathematics/ https://plato.stanford.edu/entries/nominalism-mathematics/ More interesting reading: https://philosophy.stackexchange.com/questions/81414/if-most-numbers-are-uncomputable-in-what-sense-do-they-exist https://philosophy.stackexchange.com/questions/81414/if-most... https://math.stackexchange.com/questions/4322297/in-what-sense-does-a-number-exist-if-it-is-proven-to-be-uncomputable https://math.stackexchange.com/questions/4322297/in-what-sen... Many automated theorem provers can only prove things by construction, thus computability is the requirement for 'existence' in these systems (calculus of constructions via Coq, LEAN, etc). In other words, they follow a constructivist approach to mathematics, which is rather interesting as such approaches require us to elide a lot of 'obvious' axioms we take for granted (such as the law of excluded middle). Several things shake out of this approach such as the conclusion that all functions are continuous. (https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_mathematics) https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_...)

roywiggins 3y ago

That proof assumes that "s(n)=n+1 is clearly an injection" - ie that every natural number has a unique successor, and uses it to prove something about the size of the set. But it doesn't prove there is "always a number following another number", it assumes it. imho it's part of the definition of what the set of natural numbers even is. Peano arithmetic just has S(n) always exists and is injective as axioms.

anon291 3y ago
Correct. All descriptions of the natural numbers of which I'm aware require the assumption that one can take some object and apply some operation 'S' on that object to produce another object that is neither the original object nor any of the objects produced before by repeated applications of 'S' to some root object. Whether you simply define a number as 0 and S(x) where x is a natural number or you define natural numbers as nested sets in ZFC or as church-encoded lambda terms, etc, you assume the existence of such a transformation.