You are mistaken about the Achilles heel.
This is silly. There must always be some first principle that is unjustified. For christians, it is belief in God. More broadly and more interestingly, it is also the principle of NonContradiction. One can never make an argument against the principle, because it already relies on itself. The reason this essential principle of thought and existence must be accepted is entirely pragmatic; if one denies it, he is absurd.
It is the same for science. For 1500 odd years, nonempirical science accomplished nothing. Metaphysics accomplished nothing. It is only with the introduction of empirical science that our knowledge progresses.
You are correct about one thing, however. Induction is not sufficient for true knowledge. However, science does not and cannot seek true knowledge, precisely because it relies on induction. Science seeks a sort of practical knowledge, that is, a theory only has to be good enough to suit the available facts. If it breaks, we get a new theory. Problem solved, the end.
I am not sure why you would say that there must always be some first principle that is unjutsified. Your first principle must be your ultimate authority and you must be able to deduce all your propositions from it. However, you still need to justify the principle because if you don't, then any further proposition that you make can logically be ignored or dismissed.
That is why the first principle of a worldview must contain enough information so it can justify itself and show there are no logical incoherances. I say this because if you need to appeal to justification outside of your first principle, then it cannot be your ultimate authority and fails as a first principle.
If your first principle fails, there is no foundation for the rest of your worldview.
By the way, I do not assert God as my first principle so this is a claim on your part that just isn't true. Rather, the Biblical scriptures are my ultimate authority. Yes, I believe we all have innate knowledge of God's existance however this is also deducable from scripture.
You said, "One can never make an argument against the principle, because it already relies on itself". But if empiricism is your first principle then everything must be deducible as such. But empiricism says that all our knowledge must come from our sensory experiences via observations. But the principle itself fails because the
metaphysical idea behind empiricism cannot ironically be sensed or observed.
So this first principle cannot "rely" on itself because it fails it's own test.
You said, "It is the same for science. For 1500 odd years, nonempirical science accomplished nothing".
But what is science that doesn't base itself upon empiricism? Is there such a thing? Even if there is such a thing how do you know it accomplished nothing? Why would non-empirical science want to provide empirical observations? Isn't this a contradiction? Please make sense!
You said, "It is only with the introduction of empirical science that our knowledge progresses".
Then you have to show empirically that the
only way knowledge can progress is through empiricism. At the moment, it's just an unjustified assumption.
You said, "You are correct about one thing, however. Induction is not sufficient for true knowledge. However, science does not and cannot seek true knowledge, precisely because it relies on induction".
Yes, induction is always a fallacy. But I find it strange that earlier you said that the use of science is our only way to progress knowledge and now you claim that, and I quote, "science does not and cannot seek true knowledge". So which is it? Is science fallacious or is it not?
You said, "Science seeks a sort of practical knowledge, that is, a theory only has to be good enough to suit the available facts. If it breaks, we get a new theory"
So what is practical knowledge and is practical knowledge false knowledge (as you've already said science cannot attain true knowledge)? Doesn't practical knowledge assume empiricism? Again how do you justify such a thing?
Finally you said that it's "Problem solved, the end" but you seem to have raised more questions than have been answered! Uh oh!