Are Distinct Implications Possible with Only One Axiom?

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In summary, the conversation discusses the possibility of having distinct implications from the existence of only one axiom. It is argued that technically, all axioms can be combined into a single postulate, so every axiomatic system can be thought of as having one axiom. The answer to the question is ultimately "yes." However, there is a debate about whether this approach is meaningful, as the number of axioms does not necessarily determine the depth or complexity of a system. The focus is shifted to the implication structure and the format of theorems and axioms.
  • #1
epkid08
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Is it possible to have distinct implications from the existence of only one axiom?
 
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  • #2
Technically all axioms can be conjoined into a single postulate:

A = A1 and A2 and A3...

So every axiomatic system can be though of as having 1 axiom and the answer to your question is "Yes".
 
  • #3
I know what you mean, but wouldn't you need an axiom that allows you to "combine" the axioms into one logical statement.

Anywho let me be more specific to dodge your problem then, assume you have only one axiom, the axiom of extensionality from ZFC. Can any truly distinct implications be concluded from this axiom?
 
  • #4
epkid08 said:
I know what you mean, but wouldn't you need an axiom that allows you to "combine" the axioms into one logical statement.

Anywho let me be more specific to dodge your problem then, assume you have only one axiom, the axiom of extensionality from ZFC. Can any truly distinct implications be concluded from this axiom?

Wouldn't the one axiom simply encode all the information in a way like jambaugh has said? The definition through use of intersection is universal, it doesn't take context depending on the axiom or the system/constraints its describing.
 
  • #5
epkid08 said:
I know what you mean, but wouldn't you need an axiom that allows you to "combine" the axioms into one logical statement.

Anywho let me be more specific to dodge your problem then, assume you have only one axiom, the axiom of extensionality from ZFC. Can any truly distinct implications be concluded from this axiom?

Again this depends on what you mean (I think your question is ill posed).

Suppose you have a system of axioms A1, A2, and A3 from which you formulate a set of definitions and prove a theorem T.

From just A1 you can prove T' = (A2 and A3 implies T).

By the same token you can start with 0 axioms and change each theorem to the corresponding contingent theorem. e.g. T'' = (A1 and A2 and A3 implies T).

Unless you get very specific about the format of theorems and axioms, counting how many you start with is not very meaningful. The math is not in the axioms per se but in the implication structure.
 

1. What is logic?

Logic is the study of reasoning and argumentation. It involves identifying and analyzing patterns of reasoning in order to determine whether arguments are valid or invalid.

2. What are proofs?

Proofs are a series of logical steps that demonstrate the validity of an argument or claim. They are used to show that a statement or theorem is true based on a set of axioms, definitions, and previously proven statements.

3. How do I construct a logical argument?

To construct a logical argument, you must first identify and clearly state your premises (the statements or evidence that support your argument). Then, you must use deductive reasoning to draw a conclusion that logically follows from your premises.

4. What is the difference between inductive and deductive reasoning?

Inductive reasoning involves using specific observations or evidence to form a general conclusion. Deductive reasoning, on the other hand, starts with a general principle or rule and applies it to specific cases to draw a conclusion.

5. How do I know if an argument is valid?

An argument is considered valid if the conclusion logically follows from the premises. This means that if the premises are true, the conclusion must also be true. To determine the validity of an argument, you can use logical rules and techniques such as truth tables, Venn diagrams, or formal proof systems.

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