Question

Consider a sequence of theories $T_i$ such that $T_i\ne T_{i+1}$ and $T_i\subseteq T_{i+1}$. Show that $\text{Missing left or extra right}$ is not finitely axiomatisable.

Short answer

The union is not finitely axiomatisable; it requires infinitely many axioms.

Step-by-step solution

Understanding the Terms

We have a sequence of theories $T_i$ such that $T_{i}eq{T}_{i+1}$ and $T_i\subseteq T_{i+1}$. The union of these theories, $\cup \{T_i\mid i\in \mathcal{N}\}$, combines all the elements of each theory in the infinite sequence.

Assumptions in Context

The problem requires us to show that $\cup \{T_i\mid i\in \mathcal{N}\}$ is not finitely axiomatisable. This means it cannot be described with a finite set of axioms. We start by assuming, for contradiction, that this union is finitely axiomatisable.

Analyzing Theory Inclusion

From the given $T_i\subseteq T_{i+1}$ and $T_{i}eq{T}_{i+1}$, the sequence is strictly increasing and each theory $T_i$ adds new sentences not contained in $T_{i-1}$. Thus, there is at least one sentence added for each theory $T_i$.

Contradiction Argument

Assuming finitely axiomatisable, the union $\cup \{T_i\mid i\in \mathcal{N}\}$ should be derivable from a finite subset. However, a finite subset cannot capture all these new, unique sentences in an infinite increasing sequence without missing some of them, contradicting finite axiomatisation.

Conclude Not Finitely Axiomatisable

Since assuming finite axiomatisation led to a contradiction, we conclude that $\cup \{T_i\mid i\in \mathcal{N}\}$ cannot be axiomatised with a finite number of axioms, meaning it is not finitely axiomatisable.

Key concepts

Increasing Theories

An increasing theory can be thought of as a sequence of theories where each theory includes everything in the previous theory, plus something extra. Formally, if we have a sequence of theories $T_i$, this means that $T_i\subseteq T_{i+1}$ and each $T_{i}eq{T}_{i+1}$.

This ensures that every new theory in the sequence has more sentences than its predecessor.
- Every consecutive theory adds at least one new element. - Theories are arranged in a way that makes each subsequent theory more comprehensive than the last. Increasing theories help us understand complex systems by building on previous knowledge. They highlight how theories can evolve by continually expanding and refining what they encompass. This progressive method shows the natural growth seen in mathematical and logical systems.

Infinite Sequences

Infinite sequences are an essential part of understanding increasing theories. They refer to a list of elements that go on forever. In the context of sequences of theories $T_i$, an infinite sequence means there's no final theory; they continue indefinitely.

The infinite nature of these sequences is crucial for many logical arguments:
- When something is infinite, it implies that we can keep adding more elements forever. - No matter how far you go in the sequence, you can always move to another new theory.
Infinite sequences allow us to examine concepts and problems that do not have boundaries, offering insights into fields like calculus and decision theory where processes or behaviors do not end. They are challenging to contend with since typical finite tools and techniques might not apply.

Finite Axioms

Finite axioms refer to a limited set of foundational truths or rules from which a theory can be derived. A theory is finitely axiomatisable if it is possible to determine all its truths from this small, finite set of axioms.

For example, in geometry, Euclid's axioms are a set of finite principles from which the rest of the discipline can be developed.
- Finite axioms are desirable because they provide a clear and manageable foundation. - If a theory is finitely axiomatisable, it indicates that all truths within the theory can be traced back to a limited number of starting points.
However, in a sequence of increasing theories, particularly an infinite sequence, it often becomes challenging or impossible to capture every truth with finite axioms. In such cases, as we examine with $\cup \{T_i\mid i\in \mathcal{N}\}$, the need for infinitely many axioms arises due to the continual introduction of new, unique elements.

Logical Consistency

Logical consistency is a fundamental concept that ensures no contradictions can be derived from a set of axioms within a theory. A theory is considered logically consistent if it is impossible to derive a statement and its negation from the theory's axioms.

Ensuring logical consistency is crucial for confidence in any logical or mathematical framework: - Without consistency, theories could make contradictory predictions or conclusions. - Logical consistency provides stability and reliability in logical systems and mathematical proofs.
In dealing with infinite sequences of increasing theories, maintaining logical consistency is a challenge. Each addition must be made with care to avoid contradictions with existing statements. While increasing theories continue to expand, interested in our scenario, logical consistency must be preserved to retain the theory's validity and usefulness.