Dedekind Cut, as stated by Richard Dedekind

[MidgetDwarf]

Greetings. I was wondering if anyone knew who gets the credit for the modern treatment of Dedekind cuts using what are commonly called lower cuts or upper cuts. Since one can show that a lower/ upper cut characterizes the other, so we can just work freely with either lower or upper cuts, and show that that everything we proved using lower/upper holds for the other.

Moreover, does anyone know of a paper or link having Dedekind's original formulation in modern mathematical language? I have not been able to find a source for the above. I wanted to give a presentation to a local math club whose students have just begun proof writing, and thought this was a neat a neat activity for them to familiarize themselves with sets (proofs involving sets), inequalities, in general avoidance of circular reasoning. Ie., the proof of the Dedekind cut corresponding to the square root of 2, where it is a common to see a circular reasoning from those not experienced in proof.

 

[jedishrfu]

Your difficulty in finding any paper by Dedekind may be due to it being named in his honor as opposed to him having discovered it.

https://en.wikipedia.org/wiki/Dedekind_cut

Wiki mentions Bertrand so he might a good lead.

in any event, reference 3 in the wiki article mentions Dedekind and his cut.

 

[MidgetDwarf]

Your difficulty in finding any paper by Dedekind may be due to it being named in his honor as opposed to him having discovered it.

https://en.wikipedia.org/wiki/Dedekind_cut

Wiki mentions Bertrand so he might a good lead.

in any event, reference 3 in the wiki article mentions Dedekind and his cut.

Thank you. I somehow had the Dover translation go that reference in my personal library, but never knew it. Upon clicking the reference, I told myself it looked familiar. I will update this post with further information I found useful. I found three other sources that may be of interest to others.

 

[Office_Shredder]

Bertrand worked on the idea but Dedekind also published some stuff about Dedekind cuts. Look for Continuity and Irrational Numbers.

 

[MidgetDwarf]

Bertrand worked on the idea but Dedekind also published some stuff about Dedekind cuts. Look for Continuity and Irrational Numbers.

From what I had gathered ( I can be wrong), is that the idea of Dedekind cuts (which are presented in books nowadays) is in the spirit of Bertrand.

Moreover, for an understanding of number consider Carl Boyer: A History Of Mathematics. To get a glimpse of how different cultures throughout the centuries approached the concept of what a number is, what numbers were known to them, and which ones they ignored or gave little importance too. Book 5 of Euclid (the one usually attributed to Exodus) talks about the Theory of Proportions. Now read that, then compare what is found in Continuity and Irrational numbers with Exodus's Theory Of Proportions.

 

[Stephen Tashi]

https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1008&context=rhumj