I think the question (stated in the title) is self explanatory, but here is the complete question:
Consider the triple: $\langle S, \mathscr S, Pr \rangle$
Axiom 1: $Pr(S)=2$, $Pr(\emptyset)=0$
Axiom 2: $Pr(s)+Pr(s^c)=2$ for each $s\in \mathscr S$.
Basically, this is all I am asking: if we push the bound up (for the first axiom), and likewise tweak the second axiom will we have the same Probability Theory we have right now (perhaps with slight numeric differences, but consistent nonetheless)?
Further Explanation and Context:
Given a coin with even density, we know that the probability of landing $\{H\}$ or $\{T\}$ is $\frac{1}{2}$. Now, if we modify axiom 1, as suggested, $Pr\{H\}$ would be $1$, likewise for $\{T\}$; however, modifying the second axiom will, fortunately, help us reformulate conditional Probability $Pr\{H\}$ or $Pr\{T\}$ given we land either $\{H\}$ or $\{T\}$, and we get $Pr\{H\}=\frac{1}{2}$ and $Pr\{T\}=\frac12$ given we land either $\{H\}$ or $\{T\}$.
Gae. S. rightfully pointed out that for countable and uncountable sets the measure might stop being $\sigma-$finite.
This is the essence of my question: Most "physical" spaces are finite; will such a measure, axiomatized by what was stated, above be sufficient? My guess is that it will work out just fine, but I am not quite sure if I can prove this.
Note:
I found these two questions to be similar to what I am asking, but I don't think they were helpful to me.
Best Answer
Every statement in the usual probability theory of the space $(S, \mathscr{S}, \mathbb{P})$ can be changed into a statement in your alternative theory of the space $(S,\mathscr{S},Pr)$ by changing $\mathbb{P} \mapsto \frac{Pr}{2}$. Similarly every statement in your alternative theory becomes a statement of the usual theory by $Pr \mapsto 2\mathbb{P}$. In other words, they are equivalent and one is consistent if and only if the other is.
That said, changing $1$ to $2$, in my mind, just adds needless complication. When dealing with products, for instance, the usual theory holds that $$\mathbb{P}_{S_1\times S_2}(A_1\times A_2) = (\mathbb{P}_{S_1}\otimes \mathbb{P}_{S_2})(A_1\times A_2) = \mathbb{P}_{S_1}(A_1)\mathbb{P}_{S_2}(A_2)$$ which extends how we compute area as length times width. In your theory, the equivalent statement would be $$Pr_{S_1\times S_2}(A_1\times A_2) = \frac{1}{2} Pr_{S_1}(A_1)Pr_{S_2}(A_2)$$ and it becomes worse with higher powers $$Pr_{\prod_{i=1}^n S_i}\left(\prod_{i=1}^nA_i\right) = \frac{1}{2^{n-1}}\prod_{i=1}^nPr_{S_i}(A_i),$$ which directly complicates every statement involving independence, most likely with mixed powers of $2$.
No doubt there's a philosophical razor we could use here to reject your alternative theory as being more complicated with no worthwhile payoff.
To create an alternative theory that may result in an actual benefit, you would need to do something more drastic like taking as an axiom $\widetilde{Pr}(S) = x$ is a fixed variable and work in the ring $\mathbb{R}[x,x^{-1}]$ or some extension thereof. This might force the theory to keep track of another form of "dimension", but I'm skeptical that the payoff, if it ever materializes, would be worth the massive headache caused.