1
vote
0answers
154 views

Is there a non-trivial consistency preserving transformation?

In ‎set ‎theory ‎"equiconsistency" (and not "consistency") ‎of ‎the ‎theories ‎is the‎ ‎main ‎part ‎of ‎researches. ‎So ‎we ‎usually ‎try ‎to ‎construct a‎ ‎new model ‎using a‎ ‎given ‎one. ‎In ‎the ...
12
votes
3answers
748 views

Where is the end of universe?

In some sense the empty set ($\emptyset$) and the global set of all sets ($G$) are the ends of the universe of mathematical objects. The world which $ZFC$ describes has an end from the bottom and is ...
22
votes
4answers
1k views

Illustrating Edward Nelson's Worldview with Nonstandard Models of Arithmetic

Mathematician Edward Nelson is known for his extreme views on the foundations of mathematics, variously described as "ultrafintism" or "strict finitism" (Nelson's preferred term), which came into the ...
11
votes
1answer
654 views

Elementary Equivalence =? Homotopy Equivalence

One of the most interesting novelties in recent foundational studies is Voevodsky's Homotopical Type Theory project (see here). Finally homotopy theory ideas have entered in a royal fashion the ...
1
vote
2answers
511 views

Intension vs. Extension: Coextensive relations in model and set theory

(originally posted at MSE as Same same but different: Coextensive relations in model and set theory, slightly modified) The official definition of a structure in model theory in its presumably most ...
9
votes
2answers
584 views

Consistent hierarchy of axiomatic systems

First of all, I am not an expert in model theory. I just want to get my personal view on the foundations of mathematics straight. I just learned in Sergey Melikhov's answer to another question ...
4
votes
5answers
1k views

Concrete models of abstract structures

Most mathematicians seem to be contented with the fact, that abstract structures cannot be directly modelled as such in a set theory without ur-elements. What seems to me the standard stance: Set ...