The existence or inexistence of a maximal element

Have you ever wondered why the real number line does not have a maximal element?Take . Define an element . Declare that is greater than any element in in . Can we do that? Surely! We’re defining it thus. In fact, does not even have to be a real number! It can just be someContinue reading “The existence or inexistence of a maximal element”


The mysterious linear bounded mapping

What exactly is a linear bounded mapping? The definition says is called a linear bounded mapping if . When you hear the word “bounded”, the first thing that strikes you is that the mappings can’t exceed a particular value. That all image points are within a finite outer covering. That, unfortunately, is not implied byContinue reading “The mysterious linear bounded mapping”

On making a choice between hypotheses

At 15:33, Peter Millican says “How can any criterion of reliable knowledge be chosen, unless we already have some reliable criterion for making that choice?” What does this actually mean? Say I have two hypotheses- A and B. One of them is true, whilst the other is false. But I don’t know which is which.Continue reading “On making a choice between hypotheses”

Euclidean rings and prime factorization

Now we will talk about the factorization of elements in Euclidean rings. On pg.146 of “Topics in Algebra” by Herstein, it says: “Let be a Euclidean ring. Then every element in is either a unit in or can be written as the product of a finite number of prime elements in .” This seems elementary.Continue reading “Euclidean rings and prime factorization”

Euclidean rings and generators of ideals

This is to address a point that has just been glazed over in “Topics in Algebra” by Herstein. In a Euclidean ring, for any two elements such that . Also, there exists a function such that . We also know that the element with the lowest d-value generates the whole ring . The proof ofContinue reading “Euclidean rings and generators of ideals”

Integral domains and characteristics

Today we shall talk about the characteristic of an integral domain, concentrating mainly on misconceptions and important points. An integral domain is a commutative ring with the property that if and , then . Hence, if , then or (or both). The characteristic of an integral domain is the lowest positive integer such that .Continue reading “Integral domains and characteristics”

Ordinals- just what exactly are they?!

If ordinals have not confused you, you haven’t really made a serious attempt to understand them. Let me illustrate this. If I have 5 fruits (all different) and 5 plates (all different), then I can bijectively map the fruits to plates. However, I arrange the fruits or plates, I can still bijectively map them. Let’sContinue reading “Ordinals- just what exactly are they?!”