Axioms and multiplication?

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Axioms and multiplication?

Postby >-) » Thu Oct 01, 2015 3:46 pm UTC

I am aware of only two sets of axioms in popular use, which are ZFC and Peano.
ZFC operates on sets and Peano on natural numbers.
Is there a set of axioms on real numbers?

I'm asking because this page seems to make a bold claim that multiplication isn't repeated addition
http://www.maa.org/external_archive/dev ... 06_08.html

The Peano axioms themselves don't define multiplication, but the wikipedia article mentions Peano augmented with addition and multiplication:

a*S(b) = a+(a*b)
a*0 = 0

If you expand the recursive expression, isn't what you get exactly repeated addition?

Now the page also talks about multiplication with non-natural numbers, and how that is incompatible with the idea of repeated addition.
The Peano axioms don't say anything about this. Which is why I'm looking for axioms on real numbers.

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Re: Axioms and multiplication?

Postby jestingrabbit » Thu Oct 01, 2015 5:07 pm UTC

To answer you first question, anything that is a complete ordered field is isomorphic to the real numbers. That makes a decent axiomatic foundation for them: the field axioms and enough extra bits and pieces to get the complete ordered field, which is normally accomplished by the ordered field axioms and the least upper bound property (ie a non empty subset of the reals with an upper bound has a least upper bound).

Regarding your second question... I'd say that multiplication is at least mostly repeated addition, that the multiplication on the reals derives quite directly from the multiplication on the naturals, via multiplication of rationals. This tells you something about what else multiplication is, which is breaking things up in to equal pieces, which is division, which is also multiplication. So its not exactly repeated addition, but its pretty close.

Of course, when you start to step out to complex numbers, or multiplying polynomials, the analogy breaks even worse. But the point that the guy was making about math education involving some kind of deception and reveal... fixing multiplications presentation is the least of your problems there. I mean, you start out with only whole positive numbers, and doing division there, and then there are fractions, and then there are negative numbers, and then there are vague statements about what pi is, the real numbers in a vague way, total disavowal of squareroots of negative numbers, and then they appear. Its just the way things are, I don't think you can avoid it.
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Re: Axioms and multiplication?

Postby Nicias » Mon Oct 05, 2015 12:44 am UTC

One thing to keep in mind, is that "3" the natural number is not the same thing as "3" the integer, which is not the same thing as "3" the rational number, which is not the same thing as "3" the real number, or "3" the complex number etc.

So while repeated addition makes sense for multiplication in the natural numbers, multiplication in the integers, rationals, reals, etc is an operation on a totally different set. So why should it have the same justification? Well, besides the fact that there is a natural isomorphism of each of these sets into a subset of the next. So while 3*3 (natural) can be thought of repeated addition, we shouldn't think of 3*3 (real) as repeated addition. Rather it is the result of a limiting process of repeated additions and subdivisions with a sign convention (multiplication by rationals).

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Re: Axioms and multiplication?

Postby Qaanol » Mon Oct 05, 2015 2:16 am UTC

jestingrabbit wrote:To answer you first question, anything that is a complete ordered Archimedean field is isomorphic to the real numbers.

Fixed that for you.

As for the topic at hand…ehh, I’d say that multiplication by a natural number is exactly repeated addition.

Multiplying a real by a natural? Just add the real to itself repeatedly.
Multiplying an infinite-dimensional complex vector by a natural? Just add the vector to itself repeatedly.
Multiplying an arbitrary group element by a natural? Just add the element to itself repeatedly.

No matter what structure you are working in, if you want to talk about multiplication by a natural number, it had damn well better be the same as repeated addition of whatever your objects are.
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Re: Axioms and multiplication?

Postby jestingrabbit » Mon Oct 05, 2015 5:00 am UTC

Nicias wrote:One thing to keep in mind, is that "3" the natural number is not the same thing as "3" the integer, which is not the same thing as "3" the rational number, which is not the same thing as "3" the real number, or "3" the complex number etc.

So while repeated addition makes sense for multiplication in the natural numbers, multiplication in the integers, rationals, reals, etc is an operation on a totally different set. So why should it have the same justification? Well, besides the fact that there is a natural isomorphism of each of these sets into a subset of the next.


I really don't think that this is right, or rather that its way too weak. The naturals are a subset of the integers are a subset of the rationals etc etc. By this I mean that the 3 in the reals is identically the 3 in the rationals. You might want to say that they have different operations defined on them, but 3 is 3. There isn't a map from the reals into the complex numbers, the reals are a subset of the complex numbers.

Qaanol wrote:
jestingrabbit wrote:To answer you first question, anything that is a complete ordered Archimedean field is isomorphic to the real numbers.

Fixed that for you.


Pretty sure you get Archimedean from complete, as it says here https://en.wikipedia.org/wiki/Construct ... c_approach
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Re: Axioms and multiplication?

Postby Nicias » Mon Oct 05, 2015 11:20 am UTC

Again, I disagree.

Under one construction, for instance, the number natural number 3 is the set { { { {} } } }. Whereas the complex number 3 is an ordered pair of equivalence classes Dedekind cuts of equivalences classes of ordered pairs of equivalences classes of ordered pairs of such objects.

Now, there is definitely an isomorophic copy of each of the smaller set of numbers inside the larger set, so in that sense they are the "same" but they are not actually the same object

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Re: Axioms and multiplication?

Postby dalcde » Mon Oct 05, 2015 12:47 pm UTC

jestingrabbit wrote:
Qaanol wrote:
jestingrabbit wrote:To answer you first question, anything that is a complete ordered Archimedean field is isomorphic to the real numbers.

Fixed that for you.


Pretty sure you get Archimedean from complete, as it says here https://en.wikipedia.org/wiki/Construct ... c_approach

Usually "complete" refers to Cauchy complete, and a complete ordered field need not be Archimedean (which you can show by taking the completion of an ordered field that does not satisfy the Archimedean property).

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Re: Axioms and multiplication?

Postby jestingrabbit » Mon Oct 05, 2015 1:27 pm UTC

dalcde wrote:Usually "complete" refers to Cauchy complete, and a complete ordered field need not be Archimedean (which you can show by taking the completion of an ordered field that does not satisfy the Archimedean property).


Define completeness however you like.

https://en.wikipedia.org/wiki/Completen ... mpleteness

They're all equivalent. I dare you to describe a complete ordered field that isn't the reals using any of the definitions for complete appearing on that page.

Nicias wrote:Again, I disagree.

Under one construction, for instance, the number natural number 3 is the set { { { {} } } }. Whereas the complex number 3 is an ordered pair of equivalence classes Dedekind cuts of equivalences classes of ordered pairs of equivalences classes of ordered pairs of such objects.

Now, there is definitely an isomorophic copy of each of the smaller set of numbers inside the larger set, so in that sense they are the "same" but they are not actually the same object


And again I disagree. I even disagree with that ugly ass construction of the complex numbers you allude to there, although that's aesthetic. We do indeed construct sets and define operations in this way. We do it to establish the existence of such algebraic objects. But there is only one real numbers, whether you achieve it with Dedekind cuts or Cauchy sequences. Or are you saying that the square root of 2 in the real numbers is not unique, that there are different square roots of 2 and different pi's and e's? And if you accept that these numbers are unique, then you must accept that the real numbers are also a unique object, and from there that the construction is not the object.

You can construct the natural numbers however you please, but 3 isn't the construction you use to establish the existence of the naturals. 3 is bigger than that.
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Re: Axioms and multiplication?

Postby Nicias » Mon Oct 05, 2015 2:01 pm UTC

jestingrabbit wrote:And again I disagree. I even disagree with that ugly ass construction of the complex numbers you allude to there, although that's aesthetic. We do indeed construct sets and define operations in this way. We do it to establish the existence of such algebraic objects. But there is only one real numbers, whether you achieve it with Dedekind cuts or Cauchy sequences. Or are you saying that the square root of 2 in the real numbers is not unique, that there are different square roots of 2 and different pi's and e's? And if you accept that these numbers are unique, then you must accept that the real numbers are also a unique object, and from there that the construction is not the object.

You can construct the natural numbers however you please, but 3 isn't the construction you use to establish the existence of the naturals. 3 is bigger than that.

I am saying that there is more than one object having the properties you associate to the (I assume you mean positive) square root of two. At the very least you could consider Dedekind cuts or Cauchy sequences. I wouldn't want to say that an equivalence class of Dedekind cuts is the same object as an equivalence class of Cauchy sequences.

Now, they are definitely isomorphic, but that is not the same.

In terms of 3 that you say is "bigger than that", how do you know if you have encountered such a 3? The only thing I can figure is perhaps the image of the natural number 3 under any (semi-ring?) isomorphism, but then wouldn't you be forced to conclude that any positive real number is "3" since you can isomorphicly embed N in R via exponentiation with any base?

You can't have it both ways. If "3" is this THING THING THING then multiplication is repeated addition. If "3" is something more abstract, then almost anything can be 3, and any intuitive rationale for multiplication goes out the window.

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Re: Axioms and multiplication?

Postby quantropy » Mon Oct 05, 2015 3:51 pm UTC

>-) wrote:The Peano axioms themselves don't define multiplication, but the wikipedia article mentions Peano augmented with addition and multiplication:
a*S(b) = a+(a*b)
a*0 = 0

The wikipedia article mentions that the original Peano axioms were second order, but can be replaced with first order axioms. This is important, because, as I see it, second order axioms have a flaw as axioms in that using them relies on intuitive knowledge of the system you are using, while use of first order axioms can be checked mechanically. The idea that you can do something n times if n is a natural number relies on your intuition. Hence it is not possible to express multiplication by n as repeating addition using first order logic.

This is why Presburger arithmetic (first order arithmetic without multiplication) is different from normal first order arithmetic with multiplication. Presburger arithmetic is complete ( :!: this has a different meaning to completeness of the reals discussed above), and so multiplication is central to Gödel's incompleteness theorem. In fact Gödel turns the argument around and manages to define the notion of 'doing something n times' in terms of multiplication.

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Re: Axioms and multiplication?

Postby Qaanol » Mon Oct 05, 2015 4:10 pm UTC

jestingrabbit wrote:Define completeness however you like.

https://en.wikipedia.org/wiki/Completen ... mpleteness

They're all equivalent. I dare you to describe a complete ordered field that isn't the reals using any of the definitions for complete appearing on that page.

Cauchy completeness, I choose you!
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Re: Axioms and multiplication?

Postby jestingrabbit » Mon Oct 05, 2015 5:44 pm UTC

So what's your metric on the rational functions? ie what's a Cauchy sequence and what isn't.

eta: Here's a proof you can get from Cauchy to LUB

https://en.wikipedia.org/wiki/Least-upp ... erty#Proof

So that would make the rational functions have the LUB, but they don't: 1 is an upper bound of n/x, what's the LUB?

I'm gonna say it again: all the completeness axioms are the same for an ordered field.

and more eta: Here's a math stackexchange which isolates the hard implication and shows you how to prove it.

http://math.stackexchange.com/questions ... al-numbers

ffs, this is "my first real analysis" stuff.
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Re: Axioms and multiplication?

Postby Nyktos » Mon Oct 05, 2015 11:24 pm UTC

jestingrabbit wrote:I'm gonna say it again: all the completeness axioms are the same for an ordered field.
The page you linked claims otherwise.
For an ordered field, Cauchy completeness is weaker than the other forms of completeness on this page. But Cauchy completeness and the Archimedean property taken together are equivalent to the others.


jestingrabbit wrote:https://en.wikipedia.org/wiki/Least-upper-bound_property#Proof
This relies on the archimedean property. Suppose S is a set of positive infinitesimals, so any positive non-infinitesimal is an upper bound. In particular suppose A1 is a positive infinitesimal and B1 is not infinitesimal. Then certainly (A1 + B1)/2 is an upper bound, since it's greater than B1/2 and hence not infinitesimal. So A2 = A1 (infinitesimal) and B2 = (A1 + B1)/2 (not infinitesimal) and the same argument applies. By induction, A1 = A2 = A3 = ... and in particular An does not converge to an upper bound (since A1 was chosen not to be an upper bound).

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Re: Axioms and multiplication?

Postby jestingrabbit » Tue Oct 06, 2015 11:20 am UTC

Fair enough. I apologise. I should have read closer.
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Re: Axioms and multiplication?

Postby Qaanol » Tue Oct 06, 2015 4:31 pm UTC

jestingrabbit wrote:So what's your metric on the rational functions? ie what's a Cauchy sequence and what isn't.

This is actually a more interesting question than might initially be apparent.

The standard definition of “Cauchy” talks about the limit of distances between all elements in the tail of a sequence. But the distance function on a metric space returns a real number as its distance, and the definition of “limit” requires that ε be a real number. Even though we are working in an ordered field, the measurements used for the purposes of being Cauchy are different from the measurements used when subtracting field elements—they have to be converted to real numbers first.
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Re: Axioms and multiplication?

Postby jestingrabbit » Wed Oct 07, 2015 3:47 pm UTC

Qaanol wrote:
jestingrabbit wrote:So what's your metric on the rational functions? ie what's a Cauchy sequence and what isn't.

This is actually a more interesting question than might initially be apparent.
I'm pretty sure its a dance we've done before though. You can define Cauchy without a metric, you really just need open sets.

You still haven't answered the question though.
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Re: Axioms and multiplication?

Postby flownt » Sun Oct 11, 2015 4:55 pm UTC

jestingrabbit wrote:So what's your metric on the rational functions? ie what's a Cauchy sequence and what isn't.


Constructing the reals from the rationals via Cauchy sequences is very similar to completing other metric spaces, but as you have noticed, because the theory of metric spaces uses real numbers pervasively, they are not the same. In fact you should just use rationals in this case for "distance" and the "epsilons".

A sketch:

A sequence S of rationals is Cauchy if for all rational epsilon>0 there exists a natural number N such that for all natural numbers n,m>N, |S(n)-S(m)|<epsilon

A sequence S of rationals converges to a rational limit L if for all rational epsilon>0 there exists a natural number N such that for all natural numbers n>N, |S(n)-L|<epsilon.

Two sequences S,T of rationals are convergence equivalent if the pointwise difference D(n)=S(n)-T(n) converges to 0

The real numbers are the set of Cauchy sequences of rationals modulo convergence equivalence.

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Re: Axioms and multiplication?

Postby jestingrabbit » Mon Oct 12, 2015 7:40 am UTC

flownt wrote:
jestingrabbit wrote:So what's your metric on the rational functions? ie what's a Cauchy sequence and what isn't.


Constructing the reals from the rationals via Cauchy sequences is very similar to completing other metric spaces, but as you have noticed, because the theory of metric spaces uses real numbers pervasively, they are not the same. In fact you should just use rationals in this case for "distance" and the "epsilons".

A sketch:

A sequence S of rationals is Cauchy if for all rational epsilon>0 there exists a natural number N such that for all natural numbers n,m>N, |S(n)-S(m)|<epsilon

A sequence S of rationals converges to a rational limit L if for all rational epsilon>0 there exists a natural number N such that for all natural numbers n>N, |S(n)-L|<epsilon.

Two sequences S,T of rationals are convergence equivalent if the pointwise difference D(n)=S(n)-T(n) converges to 0

The real numbers are the set of Cauchy sequences of rationals modulo convergence equivalence.


That answers the question for rational numbers, but the claim was made that the rational functions are complete in the sense that all Cauchy sequences of rational functions converge to a rational function ie a polynomial divided by another polynomial. But without knowing what the Cauchy sequences that are being claimed to be convergent are, its a pretty meaningless statement. We still don't have an answer Qaanol.
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Re: Axioms and multiplication?

Postby Qaanol » Mon Oct 12, 2015 5:15 pm UTC

To clarify, the rational functions are an ordered non-Archimedean field. Its *completion* (with respect to Cauchy sequences) is a complete ordered non-Archimedean field. This is in chapter one of “Counterexamples in Analysis”.
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