K-theory and sets
Ignacio Añón
ianon at latahona.com.uy
Thu Jun 2 02:40:07 EDT 2022
Harvey Friedman wrote:
>* Furthermore, at least so far, category theory hasn't even been useful
*>* for ANY of our dramatic foundational revelations, of which we have a
*>* few. The really useful tool for uncovering the dramatic foundational
*>* revelations has always been through the usual standard classical set
*>* theoretic foundations.*
Since this thread begun by considering the lost legacy of Grothendieck, it
might be interesting to note that things like K-theory and the index
theorem, were developed by Atiyah, Bott and others, in their attempt to
translate into simple topology, Grothendieck's new, foundational, algebraic
perspective on the Riemann-Roch theorem.
Both k-theory and the index theorem revolutionized and simplified modern
analysis beyond recognition. You can not ascribe the relevance of this work
to mere "mathematics", avoiding the revolutionary foundational importance
this new topological perspective has for f.o.m...
The type of set theory that best formalizes these ideas, is of a particular
vintage: it is best represented by set theorists like Cantor, Hausdorff,
Godel, Takeuti, Todercevic, and ill represented by set theorists in the
vein of Solovay, Woodin, Kunen, Dales...
Putting forward the notion that set theory is a simple, unified
foundational field vis a vis Category theory, falsifies the richness of
traditions and ideas within set theory: Kanamori's bitingly lucid pieces
show this with stark clarity...
Foundationally, different traditions within set theory differ, in their
outlook, much more so than set theory and category theory.
Let me mention, to be clear, that set theory in the best tradition, is far
superior to category theory foundationally: practitioners in category
theory seem unaware that much of what they see as innovative, was already
developed by Hausdorff and others more than a century ago...
Best
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