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The axiom of choice has many counterintuitive consequences like the Banach-Tarski paradox. The Hahn-Banach theorem is a consequence of the axiom of choice, but it is weaker.

I would like to know some counterintuitive consequences of the Hahn-Banach theorem. Can the Banach-Tarski paradox be derived from the Hahn-Banach theorem?

M.González
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  • Asaf Karagila answers affirmatively here: http://math.stackexchange.com/a/156216/43208 I found this through a quick Google search. – Todd Trimble Nov 05 '14 at 16:27
  • This question appears to be off-topic because it is about consequences of the axiom of choice and the OP did not search the standard reference. – Willie Wong Nov 05 '14 at 16:30
  • Also see here: http://matwbn.icm.edu.pl/ksiazki/fm/fm138/fm13813.pdf – Todd Trimble Nov 05 '14 at 16:31
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    Let me elaborate a bit on my vote to close above. Every time you have a question about whether two weak forms of axiom of choice imply each other, the very first thing you do is to search the consequences of AoC project. Using the string search you find that Banach-Tarski is form 309, and Hahn-Banach is form 52. Then you type into the first text box the two numbers "52, 309", hit enter, and you get a nice little mutual implication box. – Willie Wong Nov 05 '14 at 16:32
  • Sorry. Maybe I should delete the question, but I do not know how to do it. – M.González Nov 05 '14 at 16:32
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    No, I think it's a fine question; my only objection is that it's so easy to find the answer. But, not everybody knows what Willie Wong asserts is "standard", and he provides information well worth knowing. Let it stand. – Todd Trimble Nov 05 '14 at 16:33
  • The Axiom of Choice is true, but The Banach-Tarski paradox is obviously in error. This leaves only one possible question. Which basic principle of mathematics is wrong? – Joshua Nov 05 '14 at 23:42
  • In Computer Science, the Axiom of Choice is true (Cartesian product of nonempty sets is nonempty is considered trivial), but the well-ordering theorem is false because we have a counterexample: {+NaN, -NaN} has no ordering because +NaN <= -NaN and +NaN >= -NaN are both false. There is no least element. – Joshua Nov 05 '14 at 23:56
  • @WillieWong I am reasonably well-educated in set theory and related questions, and am on record as being sick of AC questions on MO. But I was not aware of this "standard reference". I now am, and I am better for the knowledge. – Theo Johnson-Freyd Nov 06 '14 at 01:42
  • @TheoJohnson-Freyd I was just parroting what I learned on MathOverflow. (Disclaimer: I am not a set theorist, and I am only occasionally interested in choice issues; on the other hand, I know Andres and believe him when he refers to it as the canonical way to check for questions of this kind.) – Willie Wong Nov 06 '14 at 08:23

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To aid future inquiries, let us record here the go-to reference kindly provided by Willie Wong in a comment above: http://consequences.emich.edu/conseq.htm. This provides a data base with a utility to search for known implications between weak forms of the axiom of choice.

Todd Trimble
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