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In the aftermath of the discoveries in foundations of mathematiC's
there was surprisingly little effect on mathematics as a whole. If
one looks at stan dard textbooks in different mathematical
disciplines, especially those closer to what is referred to as
applied mathematics, there is little trace of those developments
outside of mathematical logic and model theory. But it seems fair
to say that there is a widespread conviction that the principles
embodied in the Zermelo - Fraenkel theory with Choice (ZFC) are a
correct description of the set theoretic underpinnings of
mathematics. In most textbooks of the kind referred to above, there
is, of course, no discussion of these matters, and set theory is
assumed informally, although more advanced principles like Choice
or sometimes Replacement are often mentioned explicitly. This
implicitly fixes a point of view of the mathemat ical universe
which is at odds with the results in foundations. For example most
mathematicians still take it for granted that the real number
system is uniquely determined up to isomorphism, which is a correct
point of view as long as one does not accept to look at "unnatural"
interpretations of the membership relation."
In the aftermath of the discoveries in foundations of mathematiC's
there was surprisingly little effect on mathematics as a whole. If
one looks at stan dard textbooks in different mathematical
disciplines, especially those closer to what is referred to as
applied mathematics, there is little trace of those developments
outside of mathematical logic and model theory. But it seems fair
to say that there is a widespread conviction that the principles
embodied in the Zermelo - Fraenkel theory with Choice (ZFC) are a
correct description of the set theoretic underpinnings of
mathematics. In most textbooks of the kind referred to above, there
is, of course, no discussion of these matters, and set theory is
assumed informally, although more advanced principles like Choice
or sometimes Replacement are often mentioned explicitly. This
implicitly fixes a point of view of the mathemat ical universe
which is at odds with the results in foundations. For example most
mathematicians still take it for granted that the real number
system is uniquely determined up to isomorphism, which is a correct
point of view as long as one does not accept to look at "unnatural"
interpretations of the membership relation."
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