First: In any consistent logical system, some statements are true yet cannot
be proven within that system. For instance, there has to be a first
cause to the universe, but it might have already disappeared (Big Bang)
or is outside the observable universe (God?). The existence of a first
cause is factual yet unprovable.
Second: A logical system cannot prove its own consistency. Like a boss
evaluating his own performance, it will not be a
logical system.