Axiom of Choice: Choosing Identical Objects

In summary, the Axiom of Choice states that for any collection of pairs, one can choose one element from each pair. This is used to define a choice function. While it is not needed for finite collections, it is necessary for infinite collections, as there is no obvious way to make a function that selects one element from each pair without invoking this axiom. It does not have any direct physical consequences in experiments, as the real world always involves only finitely many things. However, it could potentially be important for constructing a basis in an infinite dimensional space. It also allows for concepts such as transfinite induction, where one can argue from Nth to N+1th without being able to specify an N-like label for each collection.
  • #1
Swamp Thing
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Finite set of indistinguishables -- is axiom of choice required?
From Wikipedia entry on the Axiom of Choice:
Bertrand Russell coined an analogy: for any (even infinite) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection; this makes it possible to directly define a choice function. For an infinite collection of pairs of socks (assumed to have no distinguishing features), there is no obvious way to make a function that selects one sock from each pair, without invoking the axiom of choice.
[1] What about a finite set of indistinguishable things (e.g. identical socks)? Do we need to invoke the axiom?

[2] Is there any physical consequence of this axiom, i.e. is there any physics experiment where the calculations to predict the result must incorporate the axiom of choice, even indirectly? Perhaps some experiment involving identical particles, say?
 
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  • #2
Swamp Thing said:
Summary:: Finite set of indistinguishables -- is axiom of choice required?

From Wikipedia entry on the Axiom of Choice:

[1] What about a finite set of indistinguishable things (e.g. identical socks)? Do we need to invoke the axiom?
No. We do not need AC in any case you can give an algorithm to make a choice. And if there are only finitely many things, just pick the first, e.g.
[2] Is there any physical consequence of this axiom, i.e. is there any physics experiment where the calculations to predict the result must incorporate the axiom of choice, even indirectly? Perhaps some experiment involving identical particles, say?
No. In the real world there are always only finitely many things involved in any experiment, even if you would count the particles. The infinities in models in field theory do not require AC either. It could only be important if we needed to construct a basis in an infinite dimensional space. I doubt that this is necessary and even if, then it probably hasn't any effects on real world measurements.
 
  • #3
fresh_42 said:
And if there are only finitely many things, just pick the first, e.g.
If the notion of "first" exists for a finite number of pairs of things, then it's not clear why it becomes invalid for an infinite number of such pairs. If the Nth pair supports the notion of "first" and "second", then the (N+1)th pair should support it as well. Ad infinitum?
 
  • #4
Swamp Thing said:
If the notion of "first" exists for a finite number of pairs of things, then it's not clear why it becomes invalid for an infinite number of such pairs. If the Nth pair supports the notion of "first" and "second", then the (N+1)th pair should support it as well. Ad infinitum?
Yes, but what about uncountably many? What is the first complex number? The examples always involve a description of a set, so we can probably choose one. But what if we can only prove existence? How to select such a solution?
 
  • #5
Ok, I think I see. The very notion of Nth and N+1th may not exist for an uncountable set of collections. Is that it?
 
  • #6
Swamp Thing said:
Ok, I think I see. The very notion of Nth and N+1th may not exist for an uncountable set of collections. Is that it?
Yes, one part of it. E.g. consider the interval ##[0,1)##. What is the last number in some ordering? Does it exist? AC has several equivalent formulations, one of it is that it does exist.
 
  • #7
So in a sense, the axiom of choice implies that you can apply the principle of mathematical induction (arguing from Nth to N+1th) without being able to actually specify an N-like label for each collection?
 
  • #8
Swamp Thing said:
So in a sense, the axiom of choice implies that you can apply the principle of mathematical induction (arguing from Nth to N+1th) without being able to actually specify an N-like label for each collection?
This is another concept, called transfinite induction. And yes, it requires the axiom of choice.
 
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Related to Axiom of Choice: Choosing Identical Objects

1. What is the Axiom of Choice?

The Axiom of Choice is a mathematical principle that states that given any collection of non-empty sets, it is possible to choose one element from each set, even if there are infinitely many sets in the collection.

2. How is the Axiom of Choice used in mathematics?

The Axiom of Choice is a fundamental tool in set theory and has many applications in other areas of mathematics, including topology, functional analysis, and abstract algebra. It allows mathematicians to make certain constructions and proofs that would not be possible without it.

3. What is the difference between choosing identical objects and choosing different objects?

When choosing identical objects, the Axiom of Choice allows us to select one object from each set without any specified criteria or method. When choosing different objects, the Axiom of Choice is not necessary as we can simply choose one object from each set using a specific rule or procedure.

4. Is the Axiom of Choice controversial?

Yes, the Axiom of Choice is a highly debated topic in mathematics. Some mathematicians argue that it is a necessary and intuitive principle, while others believe it leads to counterintuitive and even paradoxical results. Its use is often carefully considered and its implications are still being explored.

5. Can the Axiom of Choice be proven?

No, the Axiom of Choice cannot be proven within the standard axiomatic system of set theory. It is an independent axiom, meaning that it cannot be derived from the other axioms. However, it has been shown to be consistent with the other axioms and is widely accepted as a valid principle in mathematics.

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