What are Walks and Units?

The present unit is part of the following walks

What is the Mathematical Garden?

The mathematical garden is a website providing information about mathematics on a university level. The mathematical garden has been started in 2020 and will be continuously extended.


The content of the mathematical garden is organized in walks. A walk covers a specific mathematical area, and you are invited to explore this part of mathematics by following such a walk.

For example, the first walk is entitled The Axioms of Zermelo and Fraenkel and explains the fundamental axioms of mathematics up to a formal definition of the set $\mathbb{N}_0$ of the natural numbers.


Each walk consists of a sequence of units where one unit may appear in different walks. For example, the unit The Natural Numbers and the Principle of Induction is the last unit of the walk The Axioms of Zermelo and Fraenkel and will be the first unit of the walk Numbers explaining the construction of the sets of the natural numbers, of the integers and so on.

The mathematical garden offers its contents at quite different levels of detail such that you may choose how deep you want to delve into the material:

First of all each walk starts with an introductory unit. If you just want to have an idea about the subject of the walk, it may be sufficient to read this introduction. As an example you may want to have a look at the Unit Introduction of the Walk The Axioms of Zermelo and Fraenkel.

If you are interested in more details, you are invited to have a closer look into the other units of this walk. We tried to keep each unit as self-contained as possible such that you are encouraged to do cherry picking: Just choose the unit(s) you are most interested in.

Each unit offers an introduction with an explanation of the most important definitions and theorems of the unit. Do you want to learn more including the exact definitions? Then you are invited to study the subsequent sections of the unit where you will find the exact definitions and theorems.

Historical Notes

If you are interested in the history of mathematics, you will find historical notes with information about the origin of the definitions and theorems.

The Complete Information – pdf Documents

What about the heart of mathematics, the proofs? In order to keep the web pages lucid, we did not include proofs into the presentation on the web pages. But each unit contains at its end a pdf document that you are invited to download with the complete material including the definitions, the theorems together with their proofs and the historical notes.

So, if you are interested in the whole story of a unit, we recommend you to move directly to the pdf document. Just follow the link at the beginning of the unit. The pdf document contains the complete information of the web page so you won’t miss anything.

New Versions

Note that the website and the pdf documents are regularly updated. The pdf documents have a version number such that you can easily see whether there have been some changes. A change lock giving some information about the changes can be found at the end of each unit.

A Technical Hint about the Use of Links

At the end a little technical hint: There are two sorts of links in a unit. One sort of links refers to other websites or to other units within the mathematical garden. You will come back to the current unit by using the back button of your browser.

The second sort of links refers to some text within the same unit, for example to a definition or a theorem. You will see a small red arrow on the right. Clicking on this arrow will bring you back to the text with the link.