Hi everyone,

**Disclaimer 1:** *logic and set theory are a long way from my field, so apologies in advance if I demonstrate extreme ignorance or stupidity, and please correct me if (when?) I write stupid things. But hopefully my basic meaning should be fairly clear to everyone even if I get some details wrong.*

**Disclaimer 2:** *I admit this question might be slightly subjective. But I feel it's not too subjective, and is fairly natural and interesting to most mathematicians, out of mere curiosity.*

**Framework:** Throughout, let's assume that standard ZF set theory is consistent, and take it as our basic mathematical foundation. (*I don't necessarily think this is best, but I prefer to pin down the discussion*).

We all know that Mathematics comes in several distinct flavours: e.g. you can believe or disbelieve the Continuum Hypothesis, and both points of view are (equally?) valid; they are really just matters of opinion. Thus there are at least 2 different versions. Of course we have *infinitely many* different versions: each number $m=1,2,3,\ldots$ gives a different flavour of Mathematics, given by the axiom $2^{\aleph_0} = \aleph_m$.

**Subquestion** Does the value of $m$ really matter very much? $2^{\aleph_0} = \aleph_1$ seems a particularly special case; but I find it hard to believe there'd be very much meaningful distinction (*in terms of theorems anyone would want to consider*) between the axioms $2^{\aleph_0} = \aleph_{103}$ or $2^{\aleph_0} = \aleph_{275}$, for example.

**If desired**, we could regard these different versions of Mathematics as essentially equivalent (*in a rough sense*): the axioms all look very similar, given by a single parametrisation. We could also throw in versions with $2^{\aleph_\alpha}$, etc.

**Alternatively**, we could remove these difficulties completely by not even considering cardinals beyond $\aleph_2$ or $\aleph_3$, say; (or any $\aleph_m$ with finite $m$).

It would be really amusing if we could do the following, for then we would have (at least) $2^{\aleph_0}$ different flavours of Mathematics! (*Although I suppose there might be technical difficulties with nonconstructive infinite 0,1 strings...!*) We'd have an explicit injective function $f$ from $[0,1]$ into the class of all possible versions of Mathematics!

# Main question

**Can we find** (or prove the existence of) an *infinite* sequence of axioms $A_1, A_2, A_3, \ldots$, for which *every* sequence of true/false assignments is consistent? (e.g. the infinite string 1011001110... would mean that $A_j$ is true for $j=1,3,4,7,8,9,\ldots$ and false for $j=2,5,6,10,\ldots$; we want every string to be consistent).

**If so**, can it be done with $A_1, A_2, \ldots$ all being essentially different kinds of axioms? [*maybe it's stupidly optimistic to hope for this*]. Can it be done without ever considering $\aleph_k$ for $k>3$, say (or 4, or any fixed finite number)?

**If not**, what's a reasonable known lower bound $K$ on the number of $A_1, \ldots, A_K$ which are known to exist, so that we have at least $2^K$ essentially different versions of Mathematics?

computably enumerabletheories. However, you can also consider continuum many consistent extensions of a computable consistent theory (even though you won't be able to effectively prove all of its theorems). Also, let me remark separately that for my previous comment, each $I_j$ should be independent of ZFC union $\{I_0, I_1, \ldots, I_{j-1}\}$ and not just of ZFC. $\endgroup$ – Jason Feb 17 '11 at 23:06