Do you believe in the Axiom of Choice?

In summary, the conversation is about the Axiom of Choice and its implications in mathematics. The speakers discuss the usefulness and potential drawbacks of the Axiom of Choice, as well as different theories that involve or exclude it. They also mention Zorn's Lemma and its relation to the Axiom of Choice, as well as how it can be used to prove the Law of Dichotomy. Lastly, they provide a link to a problem and its solution involving Zorn's Lemma.

Do you believe in the Axiom of Choice?

  • Yes

    Votes: 9 81.8%
  • No

    Votes: 1 9.1%
  • Undecided

    Votes: 1 9.1%

  • Total voters
    11
  • #1
mathboy
182
0
Feel free to give your reasons.

I voted yes, because too many useful theorems are thrown out the window if Axiom of Choice is rejected. I believe that these useful theorems outweigh the surprising (strange?) results that also arise from AC (e.g. every set can be well-ordered). Also, if AC is truly a failure (based on what I have no idea), then shouldn't it have failed by now, over 100 years later? I'm assuming that there has been no physical experiment available to disprove the Axiom of Choice, am I right? Will there ever be such a physical experiment?
 
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  • #2
It's not really something to "believe in". ZFC is one theory, ZF-C is another theory.
 
  • #3
Then I guess I'm asking which theory you prefer: ZFC or ZF without AC?
 
  • #4
mathboy said:
Then I guess I'm asking which theory you prefer: ZFC or ZF without AC?
Models of ZF that satisfy choice are generally simpler objects than models of ZF that do not; you can usually say more interesting things about models of ZFC.
 
  • #5
Can someone explain to me what Zorn's lemma is?
 
  • #7
QuantumGenie said:
Can someone explain to me what Zorn's lemma is?

another axiom used to prove the linear ordering of the cardinals i.e. for any two sets A and B there exists f: A->B or f:B->A , f an injection. this of course non-exclusive or. if you want i can type out the proof in my book for you.
 
  • #8
:smile:Well I would like to take a look if its not too long!
 
  • #9
ice109 said:
another axiom used to prove the linear ordering of the cardinals i.e. for any two sets A and B there exists f: A->B or f:B->A , f an injection. this of course non-exclusive or. if you want i can type out the proof in my book for you.

Proving the Law of Dichotomy is easier using the axiom of choice. The proof using Zorn's lemma is also very elegant too:
Let K be the set of all bijections from a subset of A to a subset of B. Then every totally ordered subset L of K contains an upper bound in K (the union of the bijections in L) (showing that the upper bound in K is the most crucial part). So there is a maximal function F, which you then show has either domain A or image B.


Here's a Zorn's Lemma problem that I posed earlier:
(Show that if r partially orders X, then there exists a total order relation m such that m contains r and m totally orders X.)
https://www.physicsforums.com/showthread.php?t=208395
with my full solution typed out. Anyone wanting to add to improve my solution please feel free to do so.
 
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  • #10
I've personally always preferred invoking the well-ordering theorem for my axiom of choice needs. This particular proof is especially simple this way. :smile:
 

What is the Axiom of Choice?

The Axiom of Choice is a mathematical principle that states, given any collection of sets, it is possible to choose one element from each set in the collection. In other words, it allows for the creation of a new set by selecting one element from each existing set.

Why is the Axiom of Choice important?

The Axiom of Choice is important because it allows for the creation of new mathematical objects, such as infinite sets, that would not be possible without it. It also simplifies mathematical proofs and allows for the development of new theorems.

Is the Axiom of Choice necessary in mathematics?

The Axiom of Choice is not necessary for most basic mathematical concepts, but it becomes necessary when dealing with more complex concepts such as infinite sets and topological spaces. It is also a fundamental part of many mathematical theories and is widely accepted by mathematicians.

What are some implications of the Axiom of Choice?

The Axiom of Choice has many implications, some of which are still being studied by mathematicians. One of the most well-known implications is the Banach-Tarski paradox, which states that a solid ball can be divided into a finite number of pieces and reassembled into two identical copies of the original ball.

Do all mathematicians believe in the Axiom of Choice?

No, there are some mathematicians who do not accept the Axiom of Choice, known as constructivists. They believe that mathematical objects should only exist if they can be constructed or explicitly defined. However, the majority of mathematicians accept the Axiom of Choice as a valid and useful tool in mathematics.

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