Skolem and pessimism about proof in mathematics.

Area of Science:

• Mathematical Logic
• Foundations of Mathematics

Background:

• Historical shifts in formalization and proof since Frege.
• Debates surrounding intuitionism, Hilbert's program, and Gödel's Incompleteness Theorem.

Purpose of the Study:

• To analyze the impact of theorems like Skolem-Lowenheim on formal axiomatic systems.
• To evaluate Hilbert's belief in mathematics' ability to resolve all questions.
• To explore the limits of formal reasoning in mathematics.

Main Methods:

• Historical analysis of formalization and proof.
• Examination of key theorems (Gödel, Skolem-Lowenheim) and their implications.
• Discussion of set theory axioms and their relevance to number theory.

Main Results:

• The Skolem-Lowenheim Theorem demonstrated limitations of first-order axiom systems for unique infinite models.
• Hilbert's assumption that mathematics can resolve all its questions lacks a reasonable basis.
• Many complex questions in elementary number theory are beyond current formal reasoning capabilities.

Conclusions:

• Formal axiomatic systems have inherent limitations.
• The success of mathematics relies on 'natural problems' and human reasoning, not solely formal systems.
• Acknowledging these limitations does not diminish the achievements of mathematics.