Research programme for new foundations - Discussion Thread
This thread is for discussing the ideas laid out in this preprint. Rather than local results, it sketches the outline of a research programme for renewing the foundations of mathematical logic.
Here is a snappy summary in five theses:
1. Tarskian model theory suffers from Platonistic thinking. Instead of pretending to map formal theories into non-formal objects, model theory should be reformed to focus on the provability-invariant transformations of theories into theories.
2. Gödel’s incompleteness argument proves something important, but not what its author thought it proves. The contradiction that emerges at the end of his long and complicated proof can be redirected to refute a different target among the initial assumptions.
3. There exists an underexamined theory of formal strings based around the function of concatenation. This theory is so powerful that is bids to represent all of mathematics, rivalling set theory in its universality.
4. Mathematical constructions can only prove the existence of objects a postulate has first granted them. The existence of uncountable sets is a postulate one can make, but not one that has to be made. There are equally good alternatives. Viewed from a higher perspective Cantor’s paradise looks like a funhouse sideshow.
5. There is no such thing as Set Theory (just as there is no such thing as Geometry, only different varieties of Euclidean and non-Euclidean geometries). Many of the paradoxes and abnormalities encountered in the pursuit of the one true form of Set Theory can be resolved by admitting instead a range of related but independent axioms systems, set theories in the plural.
Here is a snappy summary in five theses:
1. Tarskian model theory suffers from Platonistic thinking. Instead of pretending to map formal theories into non-formal objects, model theory should be reformed to focus on the provability-invariant transformations of theories into theories.
2. Gödel’s incompleteness argument proves something important, but not what its author thought it proves. The contradiction that emerges at the end of his long and complicated proof can be redirected to refute a different target among the initial assumptions.
3. There exists an underexamined theory of formal strings based around the function of concatenation. This theory is so powerful that is bids to represent all of mathematics, rivalling set theory in its universality.
4. Mathematical constructions can only prove the existence of objects a postulate has first granted them. The existence of uncountable sets is a postulate one can make, but not one that has to be made. There are equally good alternatives. Viewed from a higher perspective Cantor’s paradise looks like a funhouse sideshow.
5. There is no such thing as Set Theory (just as there is no such thing as Geometry, only different varieties of Euclidean and non-Euclidean geometries). Many of the paradoxes and abnormalities encountered in the pursuit of the one true form of Set Theory can be resolved by admitting instead a range of related but independent axioms systems, set theories in the plural.
Comments (1)
Why cannot Euclidean and non-Euclidean geometries be both part of Geometry? You still talk about geometry. Why cannot there be a field about questions of shape, size, relative positions and of space?
And are you thinking of some kind Multiverse of Mathematics? So anything goes... with some axiomatic system?
Paradoxes just show we haven't grasped everything in Mathematics. Paradoxes are not for to be solved, but to be understood.