Though as I mentioned above is not an axiom that is seen as "fair game" to assume like the ZFC axioms, it is just one of many axioms beyond ZFC it might be interesting to investigate the consequences of. Finne de video og anmeldelser av denne instrumentelle sangen AXIOM av The Contortionist. Now we know it's independent of ZFC, so the name 'axiom' is not so inappropriate. I would guess the reason why its "official name" doesn't have the word 'axiom' in it is mostly historical: for a good amount of time since Cantor originated it, it was viewed as a very important open question that would eventually be proven or refuted. And the necessity of the use (and also "how much choice" is used) is often scrutinized.Īs to why CH is not called an axiom, it is often called one in my experience. Although most mathematicians view it as a correct requirement about how sets should behave, due to historical controversy and its non-constructive character, it is often at least remarked upon when a result uses it. The same is true (to a much lesser extent) with the axiom of choice, which is independent of the other axioms of ZFC. And since not everyone agrees that the CH is a valid requirement about how sets should behave (and in fact most would disagree), it's convention to state explicitly that CH was an assumption, and to not view the theorem as absolutely true, but rather as a result relative to CH. The difference is that since topology is founded on set theory, when we assume CH, we get a whole bunch of new theorems about topology.
If we want to assume the CH for whatever reason, we are working with a stronger notion of set theory than just the ZFC axioms, since the ZFC axioms don't imply it, just like if we want to assume Hausdorff-ness we are working with a stronger notion of topological space than the usual three axioms. That we view a universe of sets as a background on which all the ensuing mathematics takes place invites us to frame this as an "absolute" assumption about all of mathematics, but it's really not so different in character.
The ZFC axioms are properties that a universe of sets obeys. This use isn't really in conflict with how "axiom" is used as a general assumption about our foundations, e.g. Similarly, we have axioms that describe how a topological space behaves (the axioms of topology) as well as for stronger notions like T1-space, Hausdorff space, regular space, and all of that. For example, the definition of a group is often referred to as the group axioms, the Peano axioms define how the natural numbers behave and the complete ordered field axioms define how the real numbers behave. The media could not be loaded, either because the server or network failed or because the format is not. The word "axiom" is often used as a list of properties that define a type of structure.
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