Intereting Posts

if $f: (0,\infty) \to (0,\infty)$ is a strictly decreasing then $f \circ f$ is decreasing?
Orthogonal projection on vector space of intervals
Showing a continuous functions on a compact subset of $\mathbb{R}^3$ can be uniformly approximated by polynomials
Proving Any connected subset of R is an Interval
Where are the transcendental numbers?
An example of a Lindelöf topological space which is not $\sigma$-compact
Every integer vector in $\mathbb R^n$ with integer length is part of an orthogonal basis of $\mathbb R^n$
Understanding vector projection
Theorem for Dividing Polynomials
approximating a maximum function by a differentiable function
Finding polynomial given the remainders
About a paper by Gold & Tucker (characterizing twin primes)
Question on $\Pi_{n=1}^\infty\left(1-\frac{x^a}{\pi^an^a}\right)$ and the Riemann Zeta function
Method of proof of $\sum\limits_{n=1}^{\infty}\tfrac{\coth n\pi}{n^7}=\tfrac{19}{56700}\pi^7$
Is Infinity =Undefined?

Assume we have a set of axioms $A_0$. There exists a statement that can be formulated with these axioms that cannot be proven to be true with this system. Assume we give such a statement axiomatic status and add it to $A_0$ and construct a new set of axioms $A_1$. Continue this process indefinitely. Assume we end up with a set of axioms $A_{\infty}$, that contains infinite number of axioms. Is this set effectively generated? Is this system complete? Is it consistent?

I don’t know anything about these things, apart from what can be read in Wikipedia about Gödel’s incompleteness theorems. Just a thought I had.

- Is there a category theory notion of the image of an axiom or predicate under a functor?
- Motivating implications of the axiom of choice?
- A model of geometry with the negation of Pasch’s axiom?
- What is exactly the difference between a definition and an axiom?
- Axiom of infinity: What is an inductive set?
- Defining the Complex numbers

- Where is axiom of regularity actually used?
- Axiomatization of $\mathbb{Z}$
- Why accept the axiom of infinity?
- A first order sentence such that the finite Spectrum of that sentence is the prime numbers
- Examples of statements which are true but not provable
- Do Kolmogorov's axioms really need only disjointness rather than pairwise disjointness?
- Proving Undecidability of first order logic without first proving it for arithmetic.
- Motivating implications of the axiom of choice?
- which axiom(s) are behind the Pythagorean Theorem
- What are the primitive notions of real analysis?

First, there is nothing unusual about having a theory with infinitely many axioms. This is the case for both of the two most commonly studied “foundational” theories, Peano Arithmetic and Zermelo-Fraenkel set theory. In both of these cases, what is often described as one named axiom (induction for PA, selection and replacement in ZF) are actually axiom *schemas* — that is, each is just a recipe for generating an infinity of *different* axioms by plugging different logical formulas into the schema.

First-and-a-half, remember that the Gödel procedure for producing an unprovable (and un-dis-provable) statement requires not only that the axioms is effectively generated, but also that the theory is rich enough to be “interesting”. (For example, the theory of integers and *addition* where multiplication is not mentioned is not rich enough for Gödel’s construction to work on it).

Second, if your original theory $A_0$ is effectively generated, then your $A_\infty$ will be too — there’s a mechanical, deterministic process that will eventually print every axiom in $A_\infty$.

$A_\infty$ will be consistent too, if $A_0$ is — by the “compactness” property of formal logic which says that every inconsistent system of axioms has a *finite* subsystem that’s also inconsistent. (If there’s a proof of a contradiction in the system, then because proofs are finite by definition, the proof depends on only finitely many of the axioms). Every finite subset of $A_\infty$ will be a subset of one of the $A_n$’s, and all of these are consistent by construction.

Since $A_\infty$ is effectively generated and (because it extends $A_0$) sufficiently rich, we can repeat the Gödel process on *it*, and get at Gödel sentence for $A_\infty$. Add that to $A_\infty$ to get $A_{\infty+1}$, and proceed ad nauseam. (In this context it is traditional to write $\omega$ instead of $\infty$, and there’s a theory of the necessary numbers “beyond infinity” under the name “ordinal numbers”).

- How to evaluate this improper integral $\int_{0}^{\infty}\frac{1-x}{1-x^{n}}\,dx$?
- Limits and convolution
- What are the bases $\beta$ such that a number with non-periodic expansion can be approximated with infinitely many numbers with periodic expansion?
- Number of ways to put N indistinct objects into M indistinct boxes
- bijection between prime ideals of $R_p$ and prime ideals of $R$ contained in $P$
- Are there any interesting semigroups that aren't monoids?
- Number fields with all degrees equal to a power of three
- Does $1.0000000000\cdots 1$ with an infinite number of $0$ in it exist?
- An identity involving the Bessel function of the first kind $J_0$
- Top homology of an oriented, compact, connected smooth manifold with boundary
- the generalized Liouville theorem
- Fast $L^{1}$ Convergence implies almost uniform convergence
- Ordinals definable over $L_\kappa$
- Prove that every nonzero prime ideal is maximal in $\mathbb{Z}$
- Why is $PGL(2,4)$ isomorphic to $A_5$