I find that sometimes there are many systems in which one could prove/ describe a single idea given in Mathematics. This made me wonder, is the thing we are proving actually true beyond the proof tools themself in some sense(*)? Like consider Russel's Set theory, some of the idea we could say in that about Sets still could be said when ZFC came. So, does this mean that those statements were like a truth of life more fundamental than Naive Set theory itself?
If there is some sense in what I said above, would there be a way to pre-emptively identify these fundamental truths of life other than seeing that they can be still shown when we remove any apparent contradictions in our foundations?(the second requires actually finding contradictions)
