1,121 karma · joined March 6, 2017
The reception of that article by the group in question, and their refusal to engage on the math side of it, is what led to him writing this blog post in the first place.
The macroeconomic effects of welfare programs create a society that is better for everyone to live in. Reducing the issue to a matter of personal responsibility is a reframing that allows you to completely lose sight of the big picture, and create programs that are destined to fail by not reaching many of the people they need to.
I think from a logic standpoint this also makes sense -- getting to undecidability quickly makes taking the direct route through first-order logic more appealing.
If I'm being honest, I now do feel a little bit deprived, I probably would have enjoyed the categorical view when I was learning this too.
As it turns out, further work developing on his discovered that using a recursively enumerable schema for induction rather than a second-order induction axiom gives rise to a simpler abstraction that still has all the properties that Peano actually desired, and which makes further developments in the space much easier.
Continuing to call it Peano Arithmetic is respect for the fact that the guy got it mostly right, and it took the mathematics community many more years to refine the ideas to their current point.
Is it a shame that Galois theory isn’t presented as a historical fossil and frozen to its state of development in Galois’s lifetime? I may be making a rather big assumption, but I like to think he would be proud, and so would Peano.
Enderton, “A Mathematical Introduction to Logic, 2nd Ed.”, p,203,269-270
Kleene, “Mathematical Logic”, p.206
EDIT: It seems like you're talking about Peano's original historical formulation of arithmetic? That's all well and good but it is categorically not what is meant by "Peano Arithmetic" in any modern context. I've provided two citations from pretty far apart in time editions of common logic texts (well, "Mathematical Logic" is a bit of a weird book, but Kleene is certainly an authority) and I hope that demonstrates this.
There's a lot of reasons that the theory is pretty much always discussed as a first-order theory. The biggest, of course, is that when taken as a first-order theory it fits neatly into the proof and statement of Godel's Incompleteness Theorems, but iiuc it's just generally much less useful in a model theoretic context to take it as a second order theory (to the point where I only ever saw this discussed as a historical note, not as a mathematical one).
EDIT 2: This is all a digression anyway. Both first- and second-order PA label the start of the Z-chain as 0; so any model of PA contains 0 when interpreted as a model of PA.
2 of the axioms are:
1. For all x, x*0 = 0
2. For all x, y: x*S(y) = x*y + y