I'm genuinely struggling with this. In what way are the axioms of science (by which I think you mean rationality) a subset of the axioms of (all?) religion?
I'm sure some collaboration and editing are due on that particular phrasing, but that's the general premise behind science. And, perhaps with fringe exceptions, religious people are fine with that axiom. Exhibit A is all the religious scientists throughout history and up to right now.
http://news.gallup.com/poll/21814/evolution-creationism-inte...
I think most of the contention is people talking past each other, and I think we're at risk of that happening here, too. I'll try to be careful.
How old the universe is and whether there is a creator are theological and metaphysical questions. Whether species change over time in response to changes in their environments is an empirical question. You can't disprove a theological axiom with empirical observation. The universe could have been created in the middle of my typing this post up. There's no scientific way to prove that an omnipotent being didn't set it up that way.
So the fundamentalists need to accept that a creator could have created a universe with evolution in it. And anti-fundamentalists need to acknowledge that they can only prove that the universe looks really old when we extrapolate backwards from existing evidence.
One metaphysical question is "Could an omnipotent creator create a universe with evolution?" I think the answer is clearly "yes". The people who object at this point usually object to the "Why would a creator do that?" That's an entirely different theological question, one that doesn't affect science a single bit.
The best solution I know is that you must proceed on he basis that what you perceive is accurate, otherwise you might as well give up as the alternative is that nothing is knowable or has any value. There is a congruent theological principle that a good god would not so deceive it’s creation.
I'd strongly disagree that this is the general premise - many scientists would already cut you off at "We can learn truths".
And yes, in this case the particular phrasing is essential.
EDIT: In this particular case, the choice of using "truth" or "fact" completely changes the premise. (And even then, I've already made a boatload of assumptions here on the exact definition of each word.)
Of course there's a limit to precision in a forum discussion, but yes, definitions are damn important in this context and it's perfectly possible to at least trying to aim at a basic level of precision.
To stay at this example: (natural) sciences are generally accepted to actually not be about learning (absolute, verified) truths - and that is a cornerstone of nat. science and one of the main properties distinguishing it from religion.
We can learn about singular facts and use reason to make generalizations and predictions from those facts that we can try to validate through experiment, thus leading to theories. But scientific theories are never true, they are valid. They are a model, an approximation of what we think of reality, only validated through repeated, but ultimately finite testing, and we can never be 100% sure that they aren't...hm, wrong. (I freely admit though, that it's at this point here where it starts to get hairy in the sense of your post, especially for a non-native speaker. :) )