Is the Empty Set a Valid Vector Space? A Closer Look at the Ten Axioms

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The discussion centers on whether the empty set can be considered a valid vector space. Participants argue that while the empty set is a subset of every set, it cannot satisfy all ten axioms of a vector space, particularly the existence of an additive identity. The concept of vacuous truth is highlighted, where "for all" statements hold true for the empty set, but "there exists" statements do not. Some participants suggest that the empty set can span the zero vector space, but ultimately conclude that it lacks the necessary elements to qualify as a vector space. The consensus is that the empty set, despite its mathematical properties, does not fulfill the requirements to be classified as a vector space.
  • #31
Thanks for that, I was quite incorrectly equating {0} with { }.
 
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  • #32
Adriadne said:
Oh dear, I hope as a neophyte, I'm not making a fool of myself.
OK look. Grant me that, if the empty set is a trivial subset of V, then it must, by the definition, contain the identity. So, is there no sense, in the case that the set V is a vector space, that the zero vector can be identified with the identity?

No, I won't grant you that! Saying the empty set is a subset is not the same as saying it is a subspace!
 
  • #33
Adriadne said:
Thanks for that, I was quite incorrectly equating {0} with { }.
You're welcome.

Have fun here at PF!
 
  • #34
phoenixthoth said:
Have fun here at PF!
Ha! Be careful what you say, I have a zillion half-assed questions up my sleeve!
 

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