In ArcMap, the 'Hydrology' tool under the toolbox 'Spatial Analyst' can be used to extract the stream network, as described in detail here: extract streams from DEM. One step consists of using a conditional statement to generate a raster stream network from the flow accumulation, for which the threshold is, as far as I know, arbitrary. Therefore, the stream network obtained might vary when different thresholds are applied.
However, the definition for stream orders is definite, that is, the first-order streams are perennial streams--streams that carry water throughout the year--that have no permanently flowing tributaries. Therefore, from my understanding, we can not ensure that the smallest stream extracted from a DEM using an arbitrary (flow accumulation) threshold is actually the first-order stream by definition (using here the Strahler rule).
I am not sure whether the understanding above is correct or not. Can you clarify it?