Full Idea
If the arbitrarily given axioms do not contradict each other with all their consequences, then they are true and the things defined by the axioms exist. For me this is the criterion of truth and existence.
Gist of Idea
If axioms and their implications have no contradictions, they pass my criterion of truth and existence
Source
David Hilbert (Letter to Frege 29.12.1899 [1899]), quoted by R Kaplan / E Kaplan - The Art of the Infinite 2 'Mind'
Book Reference
Kaplan,R and Kaplan,E: 'The Art of the Infinite' [Penguin 2004], p.51
A Reaction
If an axiom says something equivalent to 'fairies exist, but they are totally undetectable', this would seem to avoid contradiction with anything, and hence be true. Hilbert's idea sounds crazy to me. He developed full Formalism later.