Discussion Overview
The discussion centers on the existence of Hamel bases for vector spaces, particularly in the context of the Axiom of Choice (AC). Participants explore whether vector spaces can exist without a Hamel basis when AC is not assumed, examining implications for specific examples like ##\mathbb R_\mathbb Q##.
Discussion Character
- Debate/contested
- Technical explanation
- Conceptual clarification
Main Points Raised
- Some participants assert that every vector space has a Hamel basis, contingent on the Axiom of Choice being true.
- Others argue that if the Axiom of Choice is false, there can exist vector spaces without a Hamel basis, citing ##\mathbb R_\mathbb Q## as a potential example.
- There are claims that the exclusion of AC only affects the proof of the existence of Hamel bases, not the existence itself.
- Some participants express uncertainty about whether the absence of AC guarantees the existence of vector spaces without a basis.
- Discussion includes the equivalence of the Axiom of Choice and the existence of Hamel bases, with some questioning the implications of this equivalence.
- Participants note that while it is consistent with ZF set theory that ##\mathbb R_\mathbb Q## does not have a basis, it remains unclear whether it actually does or does not have one.
- There is a suggestion that the discussion may relate to Gödel's incompleteness theorems, though some participants believe it does not.
Areas of Agreement / Disagreement
Participants do not reach a consensus on whether vector spaces can exist without a Hamel basis when the Axiom of Choice is excluded. Multiple competing views remain regarding the implications of AC on the existence of bases in vector spaces.
Contextual Notes
Participants highlight that without the Axiom of Choice, it is challenging to define dimensions for arbitrary vector spaces and that the existence of bases may depend on the cardinality of generating subsets.