1. What are axioms? 2. What is one way that the author proposes there might be h
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Question
1. What are axioms?2. What is one way that the author proposes there might be hidden problems with the axioms which provide for the existence of infinite sets?
4. What did Henri Poincaré say about the foundations of mathematics?
5. The result that there are different sizes of infinity ultimately follows logically from what accepted axioms of set theory?
The axioms are meant to be self-evident truths that everyone agrees on. All of math is supposed to be built up, step by step, in a rock solid way from the axioms. The problem is founded on axioms the axioms evolve, like theories, and not everyone agrees on them. In the past, people have put forward axioms which seemed obvious, but then later on it turned out that in the system.
Explanation / Answer
1. Axioms are meant to be self evident truths that everyone agrees on. All of the maths is supposed to be built up, step by step in a rock solid way from the Axioms.
2.According to the author, In the past, people have put forward axioms which seemed obvious but later on it turned out that they led inconsistencies into the system. This is the hidden problem with axiom which provide for the existence of infinite sets.
4. Henry Poincare has said the following words over a centuries ago about the foundations of mathematics:
"Though the source be obscure... still the river flows on..."
5.The result that there are different sizes of infinity ultimately follows logically from Zermelo-Fraenkel Axioms of set theory.
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