5 ms·

How to show such numbers exist? Chapter 1 states they can be constructed from the real numbers and this construction is discussed in the epilog, but (I dont hav

How to show such numbers exist? Chapter 1 states they can be constructed from the real numbers and this construction is discussed in the epilog, but (I dont have enough time probably) there seem to be only some axioms in the epilogue but not a construction (or proof of existence).

pfortuny 5y ago
It is not complicated but it requires some work and more. Define “f(x)” is an infinitesimal if lim f(x)=0 when x-> 0. which is the same as saying “f(x)” is smaller in absolute value than any real number. Nothing more. The problem: you cannot divide by infinitesimals. Now you need to “create a field” from these numbers and the reals.: this requires the axiom of choice. There is also the sign problem. But in the end, “that is all you neeed”. You think of infinitesimals as “numbers smaller than any true real number”.
- andi999 5y ago
  Thanks! I know about axiom of choice (and I am fine with it), so how do you do the field construction? I am also a bit worried, if you have a field it is also a division algebra. And I thought we know that the only divisions algebras are R, C, H? Like https://math.stackexchange.com/questions/2020399/division-algebra-field https://math.stackexchange.com/questions/2020399/division-al... (The link does not 100% fit). So how do hyperreals go around that 'no-go' theorem?
  - scapp 5y ago
    The Frobenius theorem characterizes finite-dimensional associative real division algebras as being isomorphic to the reals, the complex numbers or the quaternions. The hyperreals aren't finite-dimensional as a vector space over the reals (for example, if Ɛ is an infinitesimal hyperreal, then Ɛ, Ɛ², Ɛ³, Ɛ⁴ ... are linearly independent over the reals). A simpler example of another real division algebra (and in fact, another field) is the field of rational functions with real coefficients. This field is also infinite dimensional over the reals (for example, 1, x, x², x³... are linearly independent).
    - andi999 5y ago
      Ah yes. But then if I think of infinitesimal als functions going to 0 at 0, I can have two infinitesimals (first one is constant zero for negative values other one constant zero for positive values), which multiplied give the constant zero function. How does a construction deal with that?
      - matt-noonan 5y ago
        Although the original statement about “infinitesimals being functions that vanish at 0” was stated with confidence, it is wrong. The usual construction of the hyperreals replaces real numbers with sequences of real numbers, and also introduces a nontrivial equivalence relation on the sequences, making two sequences equivalent if they agree on a “large” set of terms. The real numbers get represented by the constant sequences, infinitesimals get represented by sequences that approach 0, and infinite numbers are represented by sequences that grow without bound. The magic is in how “large set of terms” is defined. You need a “large set” relation with the property that finite sets are not large, and for any set either the set or its complement is large. Then we can resolve your question: say you had two not-always-zero sequences that multiply to give the all-zero sequence. Then the set of zero positions is large for one of those two sequences. And that means one of your sequences is equivalent to the zero sequence. The field axioms are saved!
        andi999 5y ago
        Thanks! Do you know of any source (textbook/paper) about this construction.
        matt-noonan 5y ago
        I learned it originally from Jim Henle, and iirc he had a textbook on the hyperreals (“Infinitessimal Analysis”, possibly?) This honors project has what looks like an accurate write up of the construction along with proofs of some of the main theorems: https://ideaexchange.uakron.edu/cgi/viewcontent.cgi?article=2291&context=honors_research_projects https://ideaexchange.uakron.edu/cgi/viewcontent.cgi?article=...
mkl 5y ago
Well, no numbers exist in any real sense. Integers, real numbers, complex numbers, hyperreals, etc., they're all imagined, and built from axioms. Keisler does have a companion book that goes into more of the theory of hyperreals, and its bibliography (p209) points to deeper theoretical work too: https://people.math.wisc.edu/~keisler/foundations.html https://people.math.wisc.edu/~keisler/foundations.html
- andi999 5y ago
  They are not all build from axioms but from definition (apart from some models of natural numbers, or you go back to ZFC). Like C is build from R^2, R from the powerset of Q, Q from integer^2, integer from natural numbers. Otherwise you do not easily know if your axioms are actually consistent.
  - mkl 5y ago
    Yes, but what I meant is that there are axioms at the base of this structure, so everything is ultimately built on them. Some people put axiom foundations higher up too, like axiomatic real numbers.
    - andi999 5y ago
      Yes, there are of course axioms, but if you have a rich enough set of axioms like zfc, then one should stop piling more axioms (if not absolutely necessary). So of course the question is existence in zfc
- Koshkin 5y ago
  Well, the real existence of abstractions is a pretty involved topic in philosophy of knowledge, and so you can’t simply say that, for example, integers do not ‘really’ exist. Because they indeed do, in a very important (and, in fact, obvious) sense: you need two cats to have a fight.
  - xyzzy123 5y ago
    I flip between agreeing and disagreeing with you! I'm not well qualified to comment so I expect an informed rebuttal to this will be instructive for me. So here goes: I still find myself in the "number systems are a cultural artifact" camp. We choose which details to keep and which to throw away in our abstractions. There are anumeric cultures and it's not clear to me at all that the existence of integers is obvious (beyond what humans can subitize) unless you're immersed in a culture that has impressed them upon you since early childhood. Do you have "2 cats" or actually just "this cat and that cat"? This cat is a bit bigger but that one looks meaner. Deep down I think that an insistence on the "reality" of integers versus reals (say) is purely aesthetic. (Ignoring for the moment that our conventional constructions build one from the other in a certain way).
dwheeler 5y ago
There are various ways to construct real numbers and complex numbers. One approach is given in Landau's Foundations of Analysis (a.k.a. Grundlagen der Analysis). You might want to look at the Metamath Proof Explorer materials on constructing numbers, a good starting point is here: http://us.metamath.org/mpeuni/mmcomplex.html http://us.metamath.org/mpeuni/mmcomplex.html Metamath is a general tool that lets you specify axioms and proofs, and verifies that the proofs only depend on axioms and previously proved proofs. The Metamath Proof Explorer (MPE) is a particular application of it that uses classical logic and the ZFC set theory axioms. MPE shows how to construct numbers using these axioms, then proves a set of number axioms, and from then on uses only those axioms so that the details of any particular construction are not important. What's usually more important is showing that you can construct them.
- andi999 5y ago
  Cool link. I know some construction of the real and complex numbers, I was asking about the construction of the hyperreals used in the textbook. I couldnt find info about that on the page.
  - Smaug123 5y ago
    The construction is fairly easy but requires some fairly hefty background knowledge to make formal. In brief, though: an ordinary real number can be implemented as a sequence of rational numbers, with two sequences considered to be "the same" if they eventually get, and stay, arbitrarily near to each other. A hyperreal can be implemented as a sequence of real numbers, with two sequences considered to be "the same" if they are equal at "very many places", for a certain strict and formal definition of "very many places". (Formally, fix a nonprincipal ultrafilter on the naturals; then the requirement is that the sequences be equal on a set of indices which is a member of that ultrafilter.) A standard real z can be viewed as a hyperreal, by taking that sequence to be z, z, z, …. Another equivalent representation of that same hyperreal would be the sequence 0, z, z, z, … because that's equal to the first sequence in "very many places". There's an infinite hyperreal, implemented by the sequence 1, 2, 3, …. (In fact there are tons and tons of infinite hyperreals.) You can prove that the space of hyperreals is a field, and moreover with some model theory you can show that actually a lot of structure (in fact, all first-order structure) transfers directly over to the hyperreals from the reals.
steve76 5y ago
There's operator theory. Treat a spectrum of prime numbers like we treat particles: https://en.wikipedia.org/wiki/Riemann_hypothesis#Operator_theory https://en.wikipedia.org/wiki/Riemann_hypothesis#Operator_th...
bikenaga 5y ago
Keisler wrote a monograph to accompany the textbook: https://people.math.wisc.edu/~keisler/foundations.html https://people.math.wisc.edu/~keisler/foundations.html It's a bit more than an instructor's manual. It covers the construction of the hyperreals in two ways: One (in section 1E) is relatively short, and is like the compactness theorem approach Abraham Robinson originally used in his book ("Non-standard Analysis", Princeton U. Press). The second construction uses ultraproducts (in Chapter 1*). Keisler gives a quick introduction to the logic needed to understand the ultraproduct construction, beginning with formal languages and sentential logic. So the prerequisites are there, and explained concisely but clearly. The ultraproduct construction comes up elsewhere; for applications to other areas of math, see Martin Davis's book "Applied Nonstandard Analysis" (Dover Publications). (There's a review of the books by Keisler, Davis, and Stroyan and Luxemburg: https://www.ams.org/journals/bull/1978-84-01/S0002-9904-1978-14401-2/S0002-9904-1978-14401-2.pdf https://www.ams.org/journals/bull/1978-84-01/S0002-9904-1978...). The rest of Keisler's monograph contains mostly nonstandard proofs of many of the results in the textbook. I like it a lot, and anyone who's interested in this stuff should grab the PDF. Another book that is short and moderately elementary is "Infinitesimal Calculus" by James Henle and Eugene Kleinberg, also published by Dover [which also publishes a paper version of Keisler's text, and is generally a great publishing company]. When Keisler's book came out, someone decided that it would be reviewed for the Bulletin of the AMS by Errett Bishop, who (as I recall) was a noted constructivist. You can read the review here: https://www.ams.org/journals/bull/1977-83-02/S0002-9904-1977-14264-X/S0002-9904-1977-14264-X.pdf https://www.ams.org/journals/bull/1977-83-02/S0002-9904-1977... Having a constructivist review a calc textbook that uses nonstandard analysis was probably not a great idea. The review ends with: "Now we have a calculus text that can be used to confirm their experience of mathematics as an esoteric and meaningless exercise in technique." I believe Keisler wrote a reply in a subsequent issue of BAMS, but I can't find it.
- andi999 5y ago
  I do not see how 1E shows existence. The definition of the natural extension needs the hyperreals in the first place. (reading on mobile, maybe I am missing something here)
  - 5y ago
mcguire 5y ago
A proof of existence for any number?
- andi999 5y ago
  if you assume ZF, then 1={0} and existence follows from existence of empty set.

pfortuny 5y ago

It is not complicated but it requires some work and more. Define “f(x)” is an infinitesimal if lim f(x)=0 when x-> 0. which is the same as saying “f(x)” is smaller in absolute value than any real number. Nothing more. The problem: you cannot divide by infinitesimals. Now you need to “create a field” from these numbers and the reals.: this requires the axiom of choice. There is also the sign problem. But in the end, “that is all you neeed”. You think of infinitesimals as “numbers smaller than any true real number”.

andi999 5y ago
Thanks! I know about axiom of choice (and I am fine with it), so how do you do the field construction? I am also a bit worried, if you have a field it is also a division algebra. And I thought we know that the only divisions algebras are R, C, H? Like https://math.stackexchange.com/questions/2020399/division-algebra-field https://math.stackexchange.com/questions/2020399/division-al... (The link does not 100% fit). So how do hyperreals go around that 'no-go' theorem?
- scapp 5y ago
  The Frobenius theorem characterizes finite-dimensional associative real division algebras as being isomorphic to the reals, the complex numbers or the quaternions. The hyperreals aren't finite-dimensional as a vector space over the reals (for example, if Ɛ is an infinitesimal hyperreal, then Ɛ, Ɛ², Ɛ³, Ɛ⁴ ... are linearly independent over the reals). A simpler example of another real division algebra (and in fact, another field) is the field of rational functions with real coefficients. This field is also infinite dimensional over the reals (for example, 1, x, x², x³... are linearly independent).
  - andi999 5y ago
    Ah yes. But then if I think of infinitesimal als functions going to 0 at 0, I can have two infinitesimals (first one is constant zero for negative values other one constant zero for positive values), which multiplied give the constant zero function. How does a construction deal with that?
    - matt-noonan 5y ago
      Although the original statement about “infinitesimals being functions that vanish at 0” was stated with confidence, it is wrong. The usual construction of the hyperreals replaces real numbers with sequences of real numbers, and also introduces a nontrivial equivalence relation on the sequences, making two sequences equivalent if they agree on a “large” set of terms. The real numbers get represented by the constant sequences, infinitesimals get represented by sequences that approach 0, and infinite numbers are represented by sequences that grow without bound. The magic is in how “large set of terms” is defined. You need a “large set” relation with the property that finite sets are not large, and for any set either the set or its complement is large. Then we can resolve your question: say you had two not-always-zero sequences that multiply to give the all-zero sequence. Then the set of zero positions is large for one of those two sequences. And that means one of your sequences is equivalent to the zero sequence. The field axioms are saved!
      - andi999 5y ago
        Thanks! Do you know of any source (textbook/paper) about this construction.
        matt-noonan 5y ago
        I learned it originally from Jim Henle, and iirc he had a textbook on the hyperreals (“Infinitessimal Analysis”, possibly?) This honors project has what looks like an accurate write up of the construction along with proofs of some of the main theorems: https://ideaexchange.uakron.edu/cgi/viewcontent.cgi?article=2291&context=honors_research_projects https://ideaexchange.uakron.edu/cgi/viewcontent.cgi?article=...

mkl 5y ago

Well, no numbers exist in any real sense. Integers, real numbers, complex numbers, hyperreals, etc., they're all imagined, and built from axioms. Keisler does have a companion book that goes into more of the theory of hyperreals, and its bibliography (p209) points to deeper theoretical work too: https://people.math.wisc.edu/~keisler/foundations.html https://people.math.wisc.edu/~keisler/foundations.html

andi999 5y ago
They are not all build from axioms but from definition (apart from some models of natural numbers, or you go back to ZFC). Like C is build from R^2, R from the powerset of Q, Q from integer^2, integer from natural numbers. Otherwise you do not easily know if your axioms are actually consistent.
- mkl 5y ago
  Yes, but what I meant is that there are axioms at the base of this structure, so everything is ultimately built on them. Some people put axiom foundations higher up too, like axiomatic real numbers.
  - andi999 5y ago
    Yes, there are of course axioms, but if you have a rich enough set of axioms like zfc, then one should stop piling more axioms (if not absolutely necessary). So of course the question is existence in zfc
Koshkin 5y ago
Well, the real existence of abstractions is a pretty involved topic in philosophy of knowledge, and so you can’t simply say that, for example, integers do not ‘really’ exist. Because they indeed do, in a very important (and, in fact, obvious) sense: you need two cats to have a fight.
- xyzzy123 5y ago
  I flip between agreeing and disagreeing with you! I'm not well qualified to comment so I expect an informed rebuttal to this will be instructive for me. So here goes: I still find myself in the "number systems are a cultural artifact" camp. We choose which details to keep and which to throw away in our abstractions. There are anumeric cultures and it's not clear to me at all that the existence of integers is obvious (beyond what humans can subitize) unless you're immersed in a culture that has impressed them upon you since early childhood. Do you have "2 cats" or actually just "this cat and that cat"? This cat is a bit bigger but that one looks meaner. Deep down I think that an insistence on the "reality" of integers versus reals (say) is purely aesthetic. (Ignoring for the moment that our conventional constructions build one from the other in a certain way).

dwheeler 5y ago

There are various ways to construct real numbers and complex numbers. One approach is given in Landau's Foundations of Analysis (a.k.a. Grundlagen der Analysis). You might want to look at the Metamath Proof Explorer materials on constructing numbers, a good starting point is here: http://us.metamath.org/mpeuni/mmcomplex.html http://us.metamath.org/mpeuni/mmcomplex.html Metamath is a general tool that lets you specify axioms and proofs, and verifies that the proofs only depend on axioms and previously proved proofs. The Metamath Proof Explorer (MPE) is a particular application of it that uses classical logic and the ZFC set theory axioms. MPE shows how to construct numbers using these axioms, then proves a set of number axioms, and from then on uses only those axioms so that the details of any particular construction are not important. What's usually more important is showing that you can construct them.

andi999 5y ago
Cool link. I know some construction of the real and complex numbers, I was asking about the construction of the hyperreals used in the textbook. I couldnt find info about that on the page.
- Smaug123 5y ago
  The construction is fairly easy but requires some fairly hefty background knowledge to make formal. In brief, though: an ordinary real number can be implemented as a sequence of rational numbers, with two sequences considered to be "the same" if they eventually get, and stay, arbitrarily near to each other. A hyperreal can be implemented as a sequence of real numbers, with two sequences considered to be "the same" if they are equal at "very many places", for a certain strict and formal definition of "very many places". (Formally, fix a nonprincipal ultrafilter on the naturals; then the requirement is that the sequences be equal on a set of indices which is a member of that ultrafilter.) A standard real z can be viewed as a hyperreal, by taking that sequence to be z, z, z, …. Another equivalent representation of that same hyperreal would be the sequence 0, z, z, z, … because that's equal to the first sequence in "very many places". There's an infinite hyperreal, implemented by the sequence 1, 2, 3, …. (In fact there are tons and tons of infinite hyperreals.) You can prove that the space of hyperreals is a field, and moreover with some model theory you can show that actually a lot of structure (in fact, all first-order structure) transfers directly over to the hyperreals from the reals.

steve76 5y ago

There's operator theory. Treat a spectrum of prime numbers like we treat particles: https://en.wikipedia.org/wiki/Riemann_hypothesis#Operator_theory https://en.wikipedia.org/wiki/Riemann_hypothesis#Operator_th...

bikenaga 5y ago

Keisler wrote a monograph to accompany the textbook: https://people.math.wisc.edu/~keisler/foundations.html https://people.math.wisc.edu/~keisler/foundations.html It's a bit more than an instructor's manual. It covers the construction of the hyperreals in two ways: One (in section 1E) is relatively short, and is like the compactness theorem approach Abraham Robinson originally used in his book ("Non-standard Analysis", Princeton U. Press). The second construction uses ultraproducts (in Chapter 1*). Keisler gives a quick introduction to the logic needed to understand the ultraproduct construction, beginning with formal languages and sentential logic. So the prerequisites are there, and explained concisely but clearly. The ultraproduct construction comes up elsewhere; for applications to other areas of math, see Martin Davis's book "Applied Nonstandard Analysis" (Dover Publications). (There's a review of the books by Keisler, Davis, and Stroyan and Luxemburg: https://www.ams.org/journals/bull/1978-84-01/S0002-9904-1978-14401-2/S0002-9904-1978-14401-2.pdf https://www.ams.org/journals/bull/1978-84-01/S0002-9904-1978...). The rest of Keisler's monograph contains mostly nonstandard proofs of many of the results in the textbook. I like it a lot, and anyone who's interested in this stuff should grab the PDF. Another book that is short and moderately elementary is "Infinitesimal Calculus" by James Henle and Eugene Kleinberg, also published by Dover [which also publishes a paper version of Keisler's text, and is generally a great publishing company]. When Keisler's book came out, someone decided that it would be reviewed for the Bulletin of the AMS by Errett Bishop, who (as I recall) was a noted constructivist. You can read the review here: https://www.ams.org/journals/bull/1977-83-02/S0002-9904-1977-14264-X/S0002-9904-1977-14264-X.pdf https://www.ams.org/journals/bull/1977-83-02/S0002-9904-1977... Having a constructivist review a calc textbook that uses nonstandard analysis was probably not a great idea. The review ends with: "Now we have a calculus text that can be used to confirm their experience of mathematics as an esoteric and meaningless exercise in technique." I believe Keisler wrote a reply in a subsequent issue of BAMS, but I can't find it.

andi999 5y ago
I do not see how 1E shows existence. The definition of the natural extension needs the hyperreals in the first place. (reading on mobile, maybe I am missing something here)
- 5y ago

mcguire 5y ago

A proof of existence for any number?

andi999 5y ago
if you assume ZF, then 1={0} and existence follows from existence of empty set.