But admitting the validity of this criticism, and even going so far as to question the possibility of ever devising absolutely inclusive and, at the same time, exclusive definitions, no sufficient reason is adduced for giving up all attempt at morphological analysis.
He is not prepared to exclude the great medieval pronouncements, or the modern Roman Catholic definitions, from the list of dogmas; but on the whole he prefers to keep in view " one historical species " - Loofs suggests that he ought perhaps rather to say one individual type - that greatest group of Christian dogmas which " was created by the Greek spirit upon the soil of the gospel " (Hist.
With these definitions it is now possible to prove the following six premisses applying to finite cardinal numbers, from which Peano 2 has shown that all arithmetic can be deduced i.
The definitions of the finite ordinals can be expressed without use of the corresponding cardinals, so there is no essential priority of cardinals to ordinals.
Contrasting the above definitions of number, cardinal and ordinals, with the alternative theory that number is an ultimate idea incapable of definition, we notice that our procedure exacts a greater attention, combined with a smaller credulity; for every idea, assumed as ultimate, demands a separate act of faith.