Of course, human thought having no infinite abilities, cannot
fully operate in any realistic way, but only in a way roughly
equivalent to formal reasonings developed from some foundations ;
this work of formalization can prevent the possible errors of
intuition.
But a purely formalistic view cannot hold either because
A realistic theory aims to
describe a fixed system given from an independent reality,
so that any of its ground formulas (statements) will be either
definitely true or definitely false as determined by this system
(but the truth of a nonmathematical statement may be ambiguous,
i.e. illdefined for the given real system). From this intention,
the theory will be built by providing an initial list of
formulas called axioms : that is a hopefully true
description of the intended system as currently known or guessed.
Thus, the theory will be true if all its axioms are indeed true on
the intended system. In this case, its logical consequences
(theorems, deduced from axioms) will also be true on the intended
model.
This is usually well ensured in pure maths, but may be speculative
in other fields. In realistic theories outside pure mathematics,
the intended reality is usually a contingent one among alternative
possibilities, that (in applied mathematics) are equally possible
from a purely mathematical viewpoint. If a theory (axioms list)
does not fit a specific reality that pure mathematics cannot
suffice to identify, this can be hopefully discovered by comparing
its predictions (logical consequences) with observations : the
theory is called falsifiable.
An axiomatic theory is a theory given with an axioms
list that means to define the selection of its models
(systems it describes), as the class of all systems where these
axioms are true. This may keep an unlimited diversity of models,
that remain equally real and legitimate interpretations. By this
definition of what «model» means, the truth of the axioms of the
theory is automatic on each model (it holds by definition and is
thus not questionable). All theorems (deduced from axioms) are
also true in each model.
In pure mathematics, the usual features of both possible roles of
theories (realistic and axiomatic) automatically protect them from
the risk to be «false» as long as the formal rules are respected.
For example Euclidean geometry, in its role of first physical theory, is but one in a landscape of diverse geometries that are equally legitimate for mathematics, and the real physical space is more accurately described by the nonEuclidean geometry of General Relativity. Similarly, biology is relative to a huge number of random choices silently accumulated by Nature on Earth during billions of years.
Realistic and axiomatic theories both appear in pure mathematics, in different parts of the foundations of mathematics, as will be presented in the section on the truth concepts in mathematics.Once 1.5 is read, continue with Time in model theory:
Set theory and
Foundations of mathematics (general table of contents) 

1. First
foundations of mathematics (detailed table of contents) 
Philosophical
aspects (Each subsection roughly uses those left and up) 
1.1. Introduction
to the foundations of mathematics 1.2. Variables, sets, functions and operations 
Intuitive representation and
abstraction
Platonism vs Formalism 
1.3. Form of theories: notions, objects, metaobjects 
Realistic vs. axiomatic
theories

1.4. Structures of mathematical systems 1.5. Expressions and definable structures 
⇨Time
in Model theory
The time of interpretation
The metaphor of the usual time The infinite time between models 
1.6. Logical connectives 1.7. Classes in set theory 
Truth
undefinability
The Berry paradox
Zeno's Paradox 
1.8. Binders in set theory  Time
in Set theory
Expansion of the set
theoretical universe
Can a set contain itself ? 
1.9. Quantifiers 
The relative sense of open
quantifiers
Interpretation
of classesClasses in an expanding
universe
Concrete examples 
1.10. Formalization
of set theory 1.11. Set generation principle 
Justifying the set
generation principle
Concepts of
truth in mathematics
Alternative logical
frameworks

⇨ 2. Set theory
(continued)
