Is there a correct formulation of set theory? For example, it's been proven by Gödel and Paul Cohen that the continuum hypothesis can neither be proven nor disproven in ZFC. Should we take from this that there exists a hitherto undiscovered formulation of set theory that can conclusively establish whether the continuum hypothesis is true or false?
Suppose there is an infinitely long ladder in front of me. I do not know that this ladder is infinitely long, only that it is either a very long (but finitely long) ladder, or an infinitely long ladder. What kind of evidence would I need to give me reasonable assurance (I don't need absolute certainty) that this ladder is indeed infinitely long?
I could walk a mile along the ladder and see that it still shows no signs of stopping soon. But the finitely long ladder would still be a better hypothesis in this case, because it explains the same data with a more conservative hypothesis. If I walk two miles, the finitely long hypothesis is still better for the same reasons. No matter what test I perform, the finitely long hypothesis will still better explain the results. Does this mean that, even if infinite objects exist, empirical evidence will never provide reasonable assurance that they exist?
I have one question concerning about lines in mathematics. My teacher told me that two lines of different lengths are made up of the same number of points. he told me that if we placed one above the other and join its end points and extend it they will meet at a point (for eg.) R. he told me that we can prove that by joining one point of the longer line to the shorter line and then to the point R and by continuing doing the same. If we do so we will feel that it is made up of the same number of points.
But in my view if we place one line above the other and join its end points then both the line would be slanting towards each other (because one is longer than the other). If we remove those points and the line that we joined then equals will be left because we are removing the same number of points. If we continue doing this by drawing parallel lines then both of them will meet at a point on the centre of the shorter line and if we stii continue drawing then the lines will meet at a point such that it...
So, it's my understanding that Russell and Whitehead's project of logicism in the Principia Mathematica didn't work out. I understand that two reasons for this are (1) that some of their axioms don't seem to be derivable from pure logic and (2) Gödel's incompleteness theorems. However, particularly since symbolic logic and the philosophy of mathematics are not my area, it's hard for me to see how 1 & 2 work and defeat the project.