Paradox?

Now for something completely different. I am half way through
reading "Einstein" by Walter Isaacson. At this point I should understand
relativity, although I'm not sure. Here goes: according to my
understanding of special relativity, since a measurement of distance
cannot be made outside time and measurement of time cannot be made
outside space, then all such measurements are relative and there are no
absolute scales of distance or time. In fact, the three dimensions of
space and one of time are not independent and are combined into a four-
dimensional continuum called space-time. Since there are no absolute
measures of time, as for instance imagined in classical Newtonian
physics, there can be no universal clock with a time that started at zero
as there can be no origin of the universe with a distance set to zero.
Therefore, as some friends and I postulated in 1958 when visiting
Brussels for the Exposition (but being poor students we did not have
enough money for the entrance fee) we concluded that this means that
there could be no creation beginning at time zero and point zero (we
dubbed this the "Brussels expostulation").
But, we know now that there was a "big bang." In the 1960s, Arno Penzias
and Robert Wilson measured radiation that could only be explained by such
a cataclysmic explosion at the origin of the universe. Then several
people, including Stephen Hawking, made calculations to show that
Einstein's theory of relativity was consistent with an expanding
universe. In fact, Einstein had realized this in his general theory of
relativity, that included gravity, and had devised a so-
called "cosmological constant" that was included in his equations to keep
the universe static, or balanced, since it was not then realized that the
universe is in fact expanding. Once this was discovered in 1929 by
Edward Hubble the need for such a cosmological constant seemed
unnecessary. However, later theorists have used this constant to show
that the universe is in fact delicately balanced.
So my point is that there seems to be an apparent paradox, according to
the theories of relativity on the one hand there can be no creation point
when the time was zero, but on the other hand there was a big bang and
this fits with the expanding universe. Can any physicist out there
explain this apparent contradiction to me, or am I missing something
obvious?