If we want to show that 2+2=4, then the mathematical axioms we would probably Peano’s axioms. Peano’s axioms are about natural numbers, while Euclid’s are about geometrical figures.

If we wanted to show that 2+2=1 then we could instead use the axioms of Modular arithmetic, with a modulus of 3

Either one could be correct, depending on the context and purpose of the proof, and as long as its clear to the reader which axioms we are using.

]]>You are of course correct about the [əʊ] pronunciation of /o/ (and it can be even more extreme than that, such as the Baltimorese [ɛʊ], but then we get into the issue of whether we’re discussing phonemes or precise phones. Also, for at least some of these speakers, the “proper” [o] still occurs in some instances. For example, I realize /o/ as [əʊ]*, but I realize /ol/ as [o:] My understanding is this is fairly common in the Maryland/Pennsylvania/Delaware area (which is where I grew up), I’ve never listed specifically for it but it wouldn’t surprise me to hear other English speakers realize /ol/ the same way, or maybe as [ol].

* or maybe [əʉ]…I don’t sense the diphthong moving back, but it’s always tricky to observe your own speech patterns.

]]>Modern Standard Arabic has only /i/ /a/ and /u/.

In many varieties of English, including Southern British English and many varieties of Southern American English, there is no /o/ (the GOAT vowel has become a diphthong of the [əʊ] type).

]]>