I have always understood infinite regress to mean going backwards forever. (Forever as in endlessly, not necessarily temporally). A model would be the negative integers, if we viewed them as a model of causation. -1 is "caused" by -2, -2 is caused by -3, -3 is caused by -4, and so forth. In this model, each event has an immediate cause; yet there is no first cause.
For example this is the interpretation of infinite regress in William Lane Craig's Kalam cosmological argument, as I understand it.
Now the SEP article on infinite regress has this exactly backwards:
"An infinite regress is a series of appropriately related elements with a first member but no last member, where each element leads to or generates the next in some sense."
In other words, this looks like induction. A base case and endless succession, like the Peano axioms. The article makes this explicit:
"Peano’s axioms for arithmetic, e.g., yield an infinite regress. "
As I understand it, this is entirely backwards. The article is confusing induction, which has a base case, with infinite regression, which is essentially a recursion or induction without a base case.
I am very confused. My sense is that SEP is simply entirely wrong on this matter, and that my longtime understanding is correct. It's the negative integers that represent infinite regress; and NOT the positive integers. That is: If there is a base case, it's NOT an infinite regress. It's the absence of a base case that defines infinite regress. That's what "regress" means: To go backward.
Set-theoretically, I have always thought that infinite regress is the opposite of well-foundedness. The axiom of foundation is what prevents an infinite regress of set membership. But forward chains of membership x1 ∈ x2 ∈ x3 ∈ ... are perfectly legal, as exemplified by the finite von Neumann ordinals. I know of no one who would call that an "infinite regress." SEP is just wrong.
Any insight? Is the SEP article representing a point of view that's prevalent, and in fact directly opposed to the traditional view of infnite regress as going backward without a starting point? How is it that SEP appears to have this matter entirely wrong? Or is my own understanding wrong all these years?
(Edit) -- Perhaps I didn't make my meaning clear.
My understanding is that infinite regress is a linear order, like this:
... < a4 < a3 < a2 < a1 < a0
It has a last member but no first member. An "infinite regress of causes" is said to be impossible by the Kalam cosmological argument, therefore there must be a first cause, which must be God, etc. The earth sits on a turtle which sits on another turtle which sits on another turtle, without end. It's turtles all the way down.
Contrast that with the structure given by the Peano axioms:
a0 < a1 < a2 < a3 < ...
which has a first element but no last element. There's a big honkin' turtle at the bottom, with another turtle on its back and so forth. It's turtles all the way UP.
Now SEP defines infinite regress as the second case. In my opinion this is 100% wrong. Infinite regress is the first case. The fact that there is an order anti-isomorphism between these two structures, so that the distinction amounts to a renaming and order flipping, is irrelevant. The semantics are completely different. In the SEP model there is a turtle at the bottom. In an infinite regress, it's turtles all the way down.
And of course by induction I did mean mathematical induction, not Humean induction.
Hope this clarifies the intent of my question. Which is: Isn't the SEP article wrong about this?