# How “Principia Mathematica” builds foundations

I understand that “Principia Mathematica” tries to build foundations of mathematics. For comparison ZFC achieves same task. From what I understand ZFC are axioms formalized in First Order Logic. Question is what is Principia based on and why it took so many pages to prove 2=1+1. I suspect that this is less tedious task in ZFC.

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If you’re really interested in this, it would be worth your while to take a look at the actual Principia and see what it does and how it does it. There are a few reasons why it takes so long to prove $1+1=2$.

One reason is that the Principia starts with considerably less than the ZF axioms; it starts with only five logical axioms along the lines of $p\lor p \implies p$. It does not start with any axioms about sets or relations or numbers.

Another reason is that the foundational machinery was not fully developed in 1911. In 2013, we would define an ordered pair as a certain sort of set (typically, $\langle x, y\rangle {\buildrel \text{def}\over \equiv } \{\{x\}, \{x,y\}\}$) and then define a relation as a set of ordered pairs. This approach hadn’t been invented yet at the time Whitehead and Russell were writing. So there are several chapters of Principia Mathematica that develop properties of sets, and then several more chapters that develop properties of relations which, from a 21st-century point of view, are completely redundant. Similarly there is a chapter on propositional functions of one variable, and then a very similar chapter about functions of two variables.

But the biggest reason for the length is that the Principia proves everything in extremely small steps. For example, an important precursor to the $1+1=2$ theorem is $$∗54\cdot 43.⊢((α,β∈1)⊃((α∩β=Λ)≡(α∪β∈2)))$$ which says that if $\alpha$ and $\beta$ are sets that each have one element, then they are disjoint if and only if their union has exactly two elements. The steps in the proof are tiny. They start with the previously-proved $\ast 54\cdot 26$, which states that $\{x\}\cup\{y\}$ has two elements if and only if $x\ne y$. To get from there to $\ast 54\cdot 43$ takes about twelve steps.

The hypotheses at one point of the proof include that $\alpha$ and $\beta$ are sets with one element each. They invoke a theorem to conclude that $\alpha = \{x\}$ and $\beta=\{y\}$ for some $x$ and $y$. Most treatments would gloss over this point. Then they prove, via a previous theorem $\ast51\cdot231$, that $\{x\}\cap\{y\}$ is empty. Most treatments at this point would take as proved that $\alpha\cap \beta$ is empty, but in Principia Mathematica this is a separate, explicit deduction, justified by theorem $\ast13\cdot12$, which states that if $x=y$, then a predicate $\psi$ is true of $x$ if and only if it is true of $y$.

All these tiny steps quickly add up to a lot of paper.

Finally, Principia Mathematica does not take 300 pages to only prove $1+1=2$. There is a great deal of explanation and discussion, and a great deal more is proved besides.