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The Frobenius theorem characterizes finite-dimensional associative real division algebras as being isomorphic to the reals, the complex numbers or the quaternio

by 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=...

by scapp 5y ago

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=...