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Rahman. M
Rahman. M
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In Zermelo-Fraenkel set theory without the Axiom of Choice (AC), it is consistent to say that there are models in which:

  1. there are vector spaces without a basis;
  2. the field of complex numbers $\mathbb{C}$ only has two automorphisms (identity and complex conjugation).

Before I even ask my main question, I want to ask two "pre-questions" (as I am not a logician):

  • is my formulation "in Zermelo-Fraenkel set theory without the Axiom of Choice (AC), it is consistent to say that there are models in which ..." formally correct  ? (And if not, what is a formal correct statement ?);
  • for the second statement above, what is a precise reference in which I I can find this statement  ?

Now for my main question: is it also true that it is consistent to say that there are models of ZF set theory without AC, in which every vector space over $\mathbb{C}$ has a base (or even stronger: in which dimension is well defined), and in which $\mathbb{C}$ also has precisely two field automorphisms  ?

In Zermelo-Fraenkel set theory without the Axiom of Choice (AC), it is consistent to say that there are models in which:

  1. there are vector spaces without a basis;
  2. the field of complex numbers $\mathbb{C}$ only has two automorphisms (identity and complex conjugation).

Before I even ask my main question, I want to ask two "pre-questions" (as I am not a logician):

  • is my formulation "in Zermelo-Fraenkel set theory without the Axiom of Choice (AC), it is consistent to say that there are models in which ..." formally correct  ? (And if not, what is a formal correct statement ?);
  • for the second statement above, what is a precise reference in which I can find this statement  ?

Now for my main question: is it also true that it is consistent to say that there are models of ZF set theory without AC, in which every vector space over $\mathbb{C}$ has a base (or even stronger: in which dimension is well defined), and in which $\mathbb{C}$ also has precisely two field automorphisms  ?

In Zermelo-Fraenkel set theory without the Axiom of Choice (AC), it is consistent to say that there are models in which:

  1. there are vector spaces without a basis;
  2. the field of complex numbers $\mathbb{C}$ only has two automorphisms (identity and complex conjugation).

Before I even ask my main question, I want to ask two "pre-questions" (as I am not a logician):

  • is my formulation "in Zermelo-Fraenkel set theory without the Axiom of Choice (AC), it is consistent to say that there are models in which ..." formally correct? (And if not, what is a formal correct statement ?);
  • for the second statement above, what is a precise reference in which I can find this statement?

Now for my main question: is it also true that it is consistent to say that there are models of ZF set theory without AC, in which every vector space over $\mathbb{C}$ has a base (or even stronger: in which dimension is well defined), and in which $\mathbb{C}$ also has precisely two field automorphisms?

Notice added Authoritative reference needed by THC
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THC
THC
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Automorphisms of vector spaces and the complex numbers without choice

In Zermelo-Fraenkel set theory without the Axiom of Choice (AC), it is consistent to say that there are models in which:

  1. there are vector spaces without a basis;
  2. the field of complex numbers $\mathbb{C}$ only has two automorphisms (identity and complex conjugation).

Before I even ask my main question, I want to ask two "pre-questions" (as I am not a logician):

  • is my formulation "in Zermelo-Fraenkel set theory without the Axiom of Choice (AC), it is consistent to say that there are models in which ..." formally correct ? (And if not, what is a formal correct statement ?);
  • for the second statement above, what is a precise reference in which I can find this statement ?

Now for my main question: is it also true that it is consistent to say that there are models of ZF set theory without AC, in which every vector space over $\mathbb{C}$ has a base (or even stronger: in which dimension is well defined), and in which $\mathbb{C}$ also has precisely two field automorphisms ?